Codebook design method for multiple input multiple output system and method for using the codebook

ABSTRACT

A multiple input multiple output (MIMO) communication method using a codebook is provided. The MIMO communication method may use one or more codebooks and the codebooks may change according to a transmission rank, a channel state of a user terminal, and/or a number of feedback bits.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit under 35 U.S.C. §119(e) of a U.S. Provisional Application No. 61/106,664, filed on Oct. 20, 2008, a U.S. Provisional Application No. 61/119,057, filed on Dec. 2, 2008, and a U.S. Provisional Application No. 61/141,441, filed on Dec. 30, 2008, in the United States Patent and Trademark Office, and the benefit under 35 U.S.C. §119(a) of a Korean Patent Application No. 10-2009-0081906, filed on Sep. 1, 2009, in the Korean Intellectual Property Office, the entire disclosures of which are incorporated herein by reference for all purposes.

BACKGROUND

1. Field

The following description relates to a design method of a codebook that is used in, for example, a multiple input multiple output (MIMO) communication system, and a technology to use the codebook.

2. Description of the Related Art

Researches are being conducted to provide various types of multimedia services and to support high quality and high speed of data transmission in a wireless communication environment. Technologies associated with a multiple input multiple output (MIMO) communication system using multiple channels are in rapid development.

In a MIMO communication system, a base station and terminals may use a codebook in order to securely and efficiently manage a channel environment. A particular space may be quantized into a plurality of codewords. The plurality of codewords that is generated by quantizing the particular space may be stored in the base station and the terminals. The codewords may be a vector or a matrix according to the dimension of a channel matrix.

For example, a terminal may select a matrix or a vector corresponding to channel information from matrices or vectors included in a codebook, based on a channel that is formed between the base station and the terminal. The base station may also receive the selected matrix or vector based on the codebook to thereby recognize the channel information. The selected matrix or vector may be used where the base station performs beamforming or transmits a transmission signal via multiple antennas.

SUMMARY

In one general aspect, provided herein is a multiple input multiple output (MIMO) communication system, the system comprising a terminal to feed back feedback data using a codebook, and a base station to access a memory storing the codebook, and to precode a data stream that the base station desires to transmit using the codebook.

Other features and aspects will be apparent from the following detailed description, the drawings, and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates an exemplary multiple input multiple output (MIMO) communication system.

FIG. 2 is a block diagram illustrating an exemplary configuration of a base station.

FIG. 3 is a flowchart illustrating an exemplary MIMO communication method.

FIG. 4 is a block diagram illustrating an exemplary base station and an exemplary terminal.

Throughout the drawings and the detailed description, unless otherwise described, the same drawing reference numerals will be understood to refer to the same elements, features, and structures. The relative size and depiction of these elements may be exaggerated for clarity, illustration, and convenience.

DETAILED DESCRIPTION

The following detailed description is provided to assist the reader in gaining a comprehensive understanding of the methods, apparatuses, and/or systems described herein. Accordingly, various changes, modifications, and equivalents of the systems, apparatuses, and/or methods described herein will be suggested to those of ordinary skill in the art. Also, description of well-known functions and constructions may be omitted for increased clarity and conciseness.

Hereinafter, exemplary embodiments will be described with reference to the accompanying drawings.

FIG. 1 illustrates an exemplary multiple input multiple output (MIMO) communication system. The MIMO communication system may be a closed-loop MIMO communication system or an open-loop MIMO communication system.

Referring to FIG. 1, the exemplary MIMO communication system includes a base station 110 and a plurality of users, for example, user 1, user 2, and user n_(u), represented by reference numerals 120, 130, and 140, respectively. While FIG. 1 shows an example of a multi-user MIMO communication system, it is understood that the disclosed systems, apparatuses, and/or methods may be applicable to a single user MIMO communication system. Herein, the term “closed-loop” may indicate that the users (user 1, user 2, user n_(u)) 120, 130, and 140 may feedback data to the base station 110. The feedback data may contain channel information and the base station 110 may generate a transmission signal based on the feedback data. The codebook described herein may be applicable to an open-loop MIMO communication system as well as the closed-loop MIMO communication system.

One or more antennas may be installed in the base station 110. A single antenna or a plurality of antennas may be installed in the users 120, 130, and/or 140. A channel may be formed between the base station 110 and the users 120, 130, and/or 140. Signals may be transmitted and received via each formed channel.

The base station 110 may transmit a single data stream or at least two data streams a user. For example, the base station 110 may adopt a spatial division multiplex access (SDMA) scheme or SDM scheme. The base station 110 may select a precoding matrix from codeword matrices included in a codebook and generate a transmission signal using the selected precoding matrix.

For example, the base station 110 may transmit pilot signals to the users 120, 130, and/or 140 via downlink channels. The pilot signals may be well known to the base station 110 and the users 120, 130, and/or 140.

A terminal corresponding to a user, for example, user 120, 130, or 140, may perform receiving a signal transmitted from the base station 110. The terminal may estimate a channel that is formed between the base station 110 and the terminal using a pilot signal. The terminal may select at least one matrix or vector from a codebook and feed back information associated with the selected at least one matrix or vector. The codebook may be updated according to a channel state. The codebook may be designed according to descriptions that will be made later with reference to FIGS. 2 through 4.

A channel estimator of the terminal may estimate the channel formed between the base station 110 and the terminal using the pilot signal. A terminal corresponding to a user, for example, user 120, 130, or 140 may select, as a vector, any one vector from vectors that are included in a pre-stored codebook. The terminal may select, as a codeword matrix, any one matrix from matrices that are included in the codebook.

For example, the terminal may select, as the vector or the codeword matrix, any one vector or any one matrix from 2^(B) vectors or 2^(B) matrices according to an achievable data transmission rate or a signal-to-interference and noise ratio (SINR). In this example, B denotes a number of feedback bits. A terminal may determine its own preferred transmission rank. The transmission rank may correspond to a number of data streams.

A feedback unit of the terminal may feedback to the base station 110, information associated with the selected vector or selected codeword matrix, hereinafter, referred to as channel information or feedback data. Information associated with the codeword matrix is also referred to as matrix information (PMI) or precoding matrix information (PMI). The channel information or the feedback data used herein may include channel state information, channel quality information, and/or channel direction information.

An information receiver of the base station 110 may receive channel information and/or feedback data of the users 120, 130, and/or 140, and determine a precoding matrix based on the received channel information and/or the feedback data. The base station 110 may select one or more of the users (user 1, user 2, user n_(u)) 120, 130, and 140 according to various types of selection algorithms, for example, a semi-orthogonal user selection (SUS) algorithm, a greedy user selection (GUS) algorithm, and the like.

The same codebook as the codebook that is stored in a memory the users 120, 130, and/or 140 may be pre-stored in a memory of the base station 110. The base station 110 may determine the precoding matrix based on matrices included in the pre-stored codebook using the channel information that is fed back from the users 120, 130, and/or 140. The base station 110 may determine the precoding matrix to maximize a total data transmission rate, such as a sum rate. The MIMO communication method described herein may achieve an efficient sum rate using an optimized codebook.

The base station 110 may precode data streams, for example, data streams S₁ and S_(N), based on the determined precoding matrix to generate a transmission signal. A process of generating the transmission signal by the base station 110 is referred to as “beamforming.” Beamforming is a signal processing technique used in sensor arrays for directional signal transmission or reception. This spatial selectivity is achieved by using adaptive or fixed receive/transmit beam patterns.

Generally, precoding is beamforming to support transmission in a radio system, for example, a multi-layer transmission in MIMO radio systems. Conventional beamforming considers linear single-layer precoding so that the same signal is emitted from each of the transmit antennas with appropriate weighting such that the signal power is maximized at the receiver output. When the receiver has multiple antennas, the single-layer beamforming cannot simultaneously maximize the signal level at all of the receive antennas and so precoding is used for multi-layer beamforming in order to maximize the throughput performance of a multiple receive antenna system. In precoding, the multiple streams of the signals are emitted from the transmit antennas with independent and appropriate weighting per each antenna such that the link throughput may be maximized at the receiver output. Precoding may be performed in a unitary fashion, in a semi-unitary fashion, or in a non-unitary fashion.

A channel environment between the base station 110 and the users 120, 130, and/or 140 may be variable. Where the base station 110 and the users 120, 130, and/or 140 use a fixed codebook, it may be difficult to adaptively cope with the varying channel environment. Although it will be described in detail later, the base station 110 and the users 120, 130, and/or 140 may adaptively cope with the varying channel environment to thereby update the codebook.

The base station 110 may generate a new precoding matrix using the updated codebook. For example, the base station 110 may update a previous precoding matrix to the new precoding matrix using the updated codebook.

FIG. 2 illustrates an exemplary configuration of a base station.

Referring to FIG. 2, the exemplary base station includes a layer mapping unit 210, a MIMO encoding unit 220, a precoder 230, and Z_(i) physical antennas 240.

At least one codeword for at least one user may be mapped to at least one layer. Where the dimension of codeword X is N_(C)×1, the layer mapping unit 210 may map the codeword X to the at least one layer using a matrix P with the dimension of N_(s)×N_(c). In this example, N_(s) denotes a number of layers or a number of effective antennas. Accordingly, it is possible to acquire the following Equation 1:

s=Px  (1).

The MIMO encoding unit 220 may perform space-time modulation for S using a matrix function M with the dimension of N_(s)×N_(s). The MIMO encoding unit 220 may perform space-frequency block coding, spatial multiplexing, and the like, according to a transmission rank.

The precoder 230 may precode outputs, for example, output data streams of the MIMO encoding unit 220 to generate the transmission signal to be transmitted via the physical antennas 240. The dimension or the number of outputs, for example, the data streams of the MIMO encoding unit 220, may indicate the transmission rank. The precoder 230 may generate the transmission signal using a precoding matrix U with the dimension of N_(t)×N_(s). Accordingly, it is possible to acquire the following Equation 2:

z=UM(s)  (2).

As described herein, W denotes the precoding matrix and R denotes the transmission rank or the number of effective antennas. In this example, the dimension of the precoding matrix W is N_(t)×R. Where the MIMO encoding unit 220 uses spatial multiplexing, Z may be given by the following Equation 3:

$\begin{matrix} {z = {{WB} = {{\begin{bmatrix} u_{11} & u_{1R} \\ \vdots & \vdots \\ u_{{Nt}\; 1} & u_{{Nt}\; R} \end{bmatrix}\begin{bmatrix} s_{1} \\ \vdots \\ s_{R} \end{bmatrix}}.}}} & (3) \end{matrix}$

Referring to the above Equation 3, the precoding matrix W may also be referred to as a weighting matrix. The dimension of the precoding matrix W may be determined according to the transmission rank and the number of physical antennas. For example, where the number N_(t) of physical antennas is four and the transmission rank is 2, the precoding matrix W may be given by the following Equation 4:

$\begin{matrix} {W = {\begin{bmatrix} W_{11} & W_{12} \\ W_{21} & W_{22} \\ W_{31} & W_{32} \\ W_{41} & W_{42} \end{bmatrix}.}} & (4) \end{matrix}$

Codebook Properties

The codebook used in a closed-loop MIMO communication system or an open-loop MIMO communication system may include one or more matrices and/or one or more vectors. A precoding matrix or a precoding vector may be determined based on the one or more matrices and/or the one or more vectors included in the codebook.

A first example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication where the number of physical antennas of the base station is four, is shown in the following Equation 5:

$\begin{matrix} {{{W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}},{W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}},{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}},{W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}},{W_{5} = {{{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}} = {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & {j\;} & 1 & {- j} \end{bmatrix}}}},{and}}\begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j\;,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}.}}} \end{matrix}} & (5) \end{matrix}$

In this example, a rotation matrix is

$\begin{matrix} {{U_{{rot}\;} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}},} & \; \end{matrix}$

and a quadrature phase shift keying (QPSK) discrete Fourier transform (DFT) matrix is

${{DFT} = {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}}},$

diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

16 matrices included in the 4-bit codebook for the single user MIMO communication system may be determined according to the transmission rank, as given by the following Table 1:

Transmit Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1) = W1(;,2) C_(1,2) = W1(;,1 2) C_(1,3) = W1(;,1 2 3) C_(1,4) = W1(;,1 2 3 4) 2 C_(2,1) = W1(;,3) C_(2,2) = W1(;,1 3) C_(2,3) = W1(;,1 2 4) C_(2,4) = W2(;,1 2 3 4) 3 C_(3,1) = W1(;,4) C_(3,2) = W1(;,1 4) C_(3,3) = W1(;,1 3 4) C_(3,4) = W3(;,1 2 3 4) 4 C_(4,1) = W2(;,2) C_(4,2) = W1(;,2 3) C_(4,3) = W1(;,2 3 4) C_(4,4) = W4(;,1 2 3 4) 5 C_(5,1) = W2(;,3) C_(5,2) = W1(;,2 4) C_(5,3) = W2(;,1 2 3) C_(5,4) = W5(;,1 2 3 4) 6 C_(6,1) = W2(;,4) C_(6,2) = W1(;,3 4) C_(6,3) = W2(;,1 2 4) C_(6,4) = W6(;,1 2 3 4) 7 C_(7,1) = W3(;,1) C_(7,2) = W2(;,1 3) C_(7,3) = W2(;,1 3 4) n/a 8 C_(8,1) = W4(;,1) C_(8,2) = W2(;,1 4) C_(8,3) = W2(;,2 3 4) n/a 9 C_(9,1) = W5(;,1) C_(9,2) = W2(;,2 3) C_(9,3) = W3(;,1 2 3) n/a 10 C_(10,1) = W5(;,2) C_(10,2) = W2(;,2 4) C_(10,3) = W3(;,1 3 4) n/a 11 C_(11,1) = W5(;,3) C_(11,2) = W3(;,1 3) C_(11,3) = W4(;,1 2 3) n/a 12 C_(12,1) = W5(;,4) C_(12,2) = W3(;,1 4) C_(12,3) = W4(;,1 3 4) n/a 13 C_(13,1) = W6(;,1) C_(13,2) = W4(;,1 3) C_(13,3) = W5(;,1 2 3) n/a 14 C_(14,1) = W6(;,2) C_(14,2) = W4(;,1 4) C_(14,3) = W5(;,1 3 4) n/a 15 C_(15,1) = W6(;,3) C_(15,2) = W5(;,1 3) C_(15,3) = W6(;,1 2 4) n/a 16 C_(16,1) = W6(;,4) C_(16,2) = W6(;,2 4) C_(16,3) = W6(;,2 3 4) n/a

Referring to the above Table 1, where the transmission rank is 4, the precoding matrix may be generated based on, for example, W₁(;,1 2 3 4), W₂(;,1 2 3 4), W₃(;,1 2 3 4), W₄(;,1 2 3 4), W₅(;,1 2 3 4), and W₆(;,1 2 3 4). In this example, W_(k)(;,n m o p) denotes a matrix that includes an n^(th) column vector, an m^(th) column vector, an o^(th) column vector, and a p^(th) column vector of W_(k).

Where the transmission rank is 3, the precoding matrix may be generated based on, for example, W₁(;,1 2 3), W₁(;,1 2 4), W₁(;,1 3 4), W₁(;,2 3 4), W₂(;,1 2 3), W₂(;,1 2 4), W₂(;,1 3 4), W₂(;,2 3 4), W₃(;,1 2 3), W₃(;,1 3 4), W₄(;,1 2 3), W₄(;,1 3 4), W₅(;,1 2 3), W₅(;,1 3 4), W₆(;,1 2 4), and W₆(;,2 3 4). In this example, W_(k)(;,n m o) denotes a matrix that includes the n^(th) column vector, the m^(th) column vector, and the o^(th) column vector of W_(k).

Where the transmission rank is 2, the precoding matrix may be generated based on, for example, W₁(;,1 2), W₁(;,1 3), W₁(;,1 4), W₁(;,2 3), W₁(;,2 4), W₁(;,3 4), W₂(;,1 3), W₂(;,1 4), W₂(;,2 3), W₂(;,2 4), W₃(;,1 3), W₃(;,1 4), W₄(;,1 3), W₄(;,1 4), W₅(;,1 3), and W₆(;,2 4). In this example, W_(k)(;,n m) denotes a matrix that includes the n^(th) column vector and the m^(th) column vector of W_(k).

Where the transmission rank is 1, the precoding matrix may be generated based on, for example, W₁(;,2), W₁(;,3), W₁(;,4), W₂(;,2), W₂(;,3), W₂(;,4), W₃(;,1), W₄(;,1), W₅(;,1), W₅(;,2), W₅(;,3), W₅(;,4), W₆(;,1), W₆(;,2), W₆(;,3), and W₆(;,4). In this example, W_(k)(;,n) denotes the n^(th column vector of W) _(k).

The codewords included in the above Table 1 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{6,2} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000\; }} \end{matrix}$

$C_{9,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000\; }} \end{matrix}$ $C_{10,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000\; }} \end{matrix}$

$C_{13,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {- 0.5000} \end{matrix}$ $C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{3,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$

$C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {{0 + {0.5000\; }}\;} & {0 - {0.5000}} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{7,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000\; }} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$

$C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000\; }} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000\; }} & 0.5000 \end{matrix}$ $C_{12,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}$

$C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000\; }} & 0.5000 \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000\; }} \end{matrix}$ $C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ $C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ $C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}$

$C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 & {0 - {0.5000}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

For the first example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to an exemplary embodiment may be designed by the following Equation 6:

$\begin{matrix} {{W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}}\begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}.}}} \end{matrix}} & (6) \end{matrix}$

Codewords included in the codebook for the multi-user MIMO communication system may be expressed using a transmission rank as given by the following Table 2:

codeword used at Transmit Codebook the mobile station Index for quantization 1 M₁=W3(;,1) 2 M₂=W3(;,2) 3 M₃=W3(;,3) 4 M₄=W3(;,4) 5 M₅=W6(;,1) 6 M₆=W6(;,2) 7 M₇=W6(;,3) 8 M₈=W6(;,4)

Referring to the above Table 2, where a transmission rank of each of users is one, the precoding matrix may be constructed by combining W₃(;1), W₃(;2), W₃(;3), W₃(;4), W₆(;1), W₆(;2), W₆(;3), and W₆(;4). In this example, W_(k)(;n) denotes the n^(th) column vector of W_(k).

The codewords included in the above Table 2 may be expressed equivalent as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $M_{4} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station that includes four physical antennas. Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 3. In the multi-user MIMO communication system, a rank of each of users is 1:

Transmit Codebook Index Rank 1 1 C_(1,1)=W1(:,2) 2 C_(2,1)=W1(:,3) 3 C_(3,1)=W1(:,4) 4 C_(4,1)=W2(:,2) 5 C_(5,1)=W2(:,3) 6 C_(6,1)=W2(:,4) 7 C_(7,1)=W3(:,1) 8 C_(8,1)=W4(:,1) 9 C_(9,1)=W5(:,1) 10 C_(10,1)=W5(:,2) 11 C_(11,1)=W5(:,3) 12 C_(12,1)=W5(:,4) 13 C_(13,1)=W6(:,1) 14 C_(14,1)=W6(:,2) 15 C_(15,1)=W6(:,3) 16 C_(16,1)=W6(:,4)

The codewords included in the above Table 3 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

Illustrated below is a second example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication system where a number of physical antennas of a base station is four.

In this example, a rotation matrix U_(rot) and a QPSK DFT matrix may be defined as follows:

$U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}$ and ${DFT} = {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}.}}$

Also, diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix} \begin{matrix} {W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{5} = {{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & j & 1 & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

When the number of physical antennas of the base station is four, matrices or codewords included in the codebook used in the single user MIMO communication system according to the second example may be given by the following Table 4:

Transmit Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1)=W1(:,2) C_(1,2)=W1(:,2 1) C_(1,3)=W1(:,1 2 3) C_(1,4)=W1(:,1 2 3 4) 2 C_(2,1)=W1(:,3) C_(2,2)=W1(:,3 1) C_(2,3)=W1(:,1 2 4) C_(2,4)=W2(:,1 2 3 4) 3 C_(3,1)=W1(:,4) C_(3,2)=W1(:,4 1) C_(3,3)=W1(:,1 3 4) C_(3,4)=W3(:,1 2 3 4) 4 C_(4,1)=W2(:,2) C_(4,2)=W1(:,2 3) C_(4,3)=W1(:,2 3 4) C_(4,4)=W4(:,1 2 3 4) 5 C_(5,1)=W2(:,3) C_(5,2)=W1(:,2 4) C_(5,3)=W2(:,1 2 3) C_(5,4)=W5(:,1 2 3 4) 6 C_(6,1)=W2(:,4) C_(6,2)=W1(:,3 4) C_(6,3)=W2(:,1 2 4) C_(6,4)=W6(:,1 2 3 4) 7 C_(7,1)=W3(:,1) C_(7,2)=W2(:,3 1) C_(7,3)=W2(:,1 3 4) n/a 8 C_(8,1)=W4(:,1) C_(8,2)=W2(:,4 1) C_(8,3)=W2(:,2 3 4) n/a 9 C_(9,1)=W5(:,1) C_(9,2)=W2(:,2 3) C_(9,3)=W5(:,1 2 3) n/a 10 C_(10,1)=W5(:,2) C_(10,2)=W2(:,2 4) C_(10,3)=W5(:,1 2 4) n/a 11 C_(11,1)=W5(:,3) C_(11,2)=W3(:,3 1) C_(11,3)=W5(:,1 3 4) n/a 12 C_(12,1)=W5(:,4) C_(12,2)=W3(:,4 1) C_(12,3)=W5(:,2 3 4) n/a 13 C_(13,1)=W6(:,1) C_(13,2)=W4(:,3 1) C_(13,3)=W6(:,1 2 3) n/a 14 C_(14,1)=W6(:,2) C_(14,2)=W4(:,4 1) C_(14,3)=W6(:,1 2 4) n/a 15 C_(15,1)=W6(:,3) C_(15,2)=W5(:,1 3) C_(15,3)=W6(:,1 3 4) n/a 16 C_(16,1)=W6(:,4) C_(16,2)=W6(:,2 4) C_(16,3)=W6(:,2 3 4) n/a

The codewords included in the above Table 4 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ {- 0.5000} \\ {- 0.5000} \end{matrix}$ $C_{3,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ {- 0.5000} \\ 0.5000 \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \end{matrix}$

$C_{5,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{6,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000}} & {{0 + {0.5000\; }}\ } \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000\; }} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000\; }} \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000\; }} \end{matrix}$

$C_{9,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000\; }} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000\; }} \end{matrix}$ $C_{10,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 - {0.5000\; }} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 + {0.5000\; }} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000}} & 0.5000 \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000}} & 0.5000 \end{matrix}$

$C_{13,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} \\ {- 0.5000} & 0.5000 \\ 0.5000 & {0 + {0.5000}} \end{matrix}$ $C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{0 + {0.5000\; }}\;} & {0 - {0.5000}} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$ $C_{7,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000\; }} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$

$C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000\; }} & 0.5000 \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000\; }} \end{matrix}$ $C_{12,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000\; }} \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & 0.5000 & {0 - {0.5000\; }} \end{matrix}$

$C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0 + {5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ $C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 & {0 - {0.5000}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

For the second example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to the second example, may be designed by combining subsets of two matrices, for example, W₃ and W₆ as follows:

$\begin{matrix} \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

When the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the multi-user MIMO communication system according to the second example may be given by the following Table 5:

Transmit Codebook Index Rank 1 1 M₁=W3(:,1) 2 M₂=W3(:,2) 3 M₃=W3(:,3) 4 M₄=W3(:,4) 5 M₅=W6(:,1) 6 M₆=W6(:,2) 7 M₇=W6(:,3) 8 M₈=W6(:,4)

The codewords included in the above Table 5 may be expressed equivalently as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $M_{4} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station including four physical antennas.

Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 6:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1)=W1(;,2) 2 C_(2,1)=W1(;,3) 3 C_(3,1)=W1(;,4) 4 C_(4,1)=W2(;,2) 5 C_(5,1)=W2(;,3) 6 C_(6,1)=W2(;,4) 7 C_(7,1)=W3(;,1) 8 C_(8,1)=W4(;,1) 9 C_(9,1)=W5(;,1) 10 C_(10,1)=W5(;,2) 11 C_(11,1)=W5(;,3) 12 C_(12,1)=W5(;,4) 13 C_(13,1)=W6(;,1) 14 C_(14,1)=W6(;,2) 15 C_(15,1)=W6(;,3) 16 C_(16,1)=W6(;,4)

The codewords included in the above Table 6 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

A third example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication system where the number of physical antennas of a base station is four is illustrated below.

In this example, a rotation matrix U_(rot) and a QPSK DFT matrix may be defined as follows.

$U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}$ and ${DFT} = {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}.}}$

Also, diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix} \begin{matrix} {W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & 1 & {- 1} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- j} & j \\ 1 & 1 & 1 & 1 \\ j & {- j} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{4} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- 1} & 1 \\ 1 & 1 & 1 & 1 \\ j & {- j} & 1 & {- 1} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{5} = {{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & j & 1 & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

For example, where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the single user MIMO communication system according to the third example may be given by the following Table 7:

Transmit Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1)=W1(:,2) C_(1,2)=W1(:,12) C_(1,3)=W1(:,1 2 3) C_(1,4)=W1(:,1 2 3 4) 2 C_(2,1)=W1(:,3) C_(2,2)=W1(:,13) C_(2,3)=W1(:,1 2 4) C_(2,4)=W2(:,1 2 3 4) 3 C_(3,1)=W1(:,4) C_(3,2)=W1(:,14) C_(3,3)=W1(:,1 3 4) C_(3,4)=W3(:,1 2 3 4) 4 C_(4,1)=W2(:,2) C_(4,2)=W1(:,23) C_(4,3)=W1(:,2 3 4) C_(4,4)=W4(:,1 2 3 4) 5 C_(5,1)=W2(:,3) C_(5,2)=W1(:,24) C_(5,3)=W2(:,1 2 3) C_(5,4)=W5(:,1 2 3 4) 6 C_(6,1)=W2(:,4) C_(6,2)=W1(:,34) C_(6,3)=W2(:,1 2 4) C_(6,4)=W6(:,1 2 3 4) 7 C_(7,1)=W3(:,1) C_(7,2)=W2(:,13) C_(7,3)=W2(:,1 3 4) n/a 8 C_(8,1)=W4(:,1) C_(8,2)=W2(:,14) C_(8,3)=W2(:,2 3 4) n/a 9 C_(9,1)=W5(:,1) C_(9,2)=W2(:,23) C_(9,3)=W5(:,1 2 3) n/a 10 C_(10,1)=W5(:,2) C_(10,2)=W2(:,24) C_(10,3)=W5(:,1 2 4) n/a 11 C_(11,1)=W5(:,3) C_(11,2)=W3(:,13) C_(11,3)=W5(:,1 3 4) n/a 12 C_(12,1)=W5(:,4) C_(12,2)=W3(:,14) C_(12,3)=W5(:,2 3 4) n/a 13 C_(13,1)=W6(:,1) C_(13,2)=W4(:,13) C_(13,3)=W6(:,1 2 3) n/a 14 C_(14,1)=W6(:,2) C_(14,2)=W4(:,14) C_(14,3)=W6(:,1 2 4) n/a 15 C_(15,1)=W6(:,3) C_(15,2)=W5(:,13) C_(15,3)=W6(:,1 3 4) n/a 16 C_(16,1)=W6(:,4) C_(16,2)=W6(:,24) C_(16,3)=W6(:,2 3 4) n/a

The codewords included in the above Table 7 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$

$C_{5.2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{6,2} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{9,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 - {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$

$C_{10,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 \\ {0 - {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000\; }} \end{matrix}$ $C_{13,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & 0.5000 \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {- 0.5000} \end{matrix}$

$C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536\; }} & {0.3536 - {0.3536\; }} \\ {0 - {0.5000\; }} & {0 - {0.5000\; }} \\ {0.3536 + {0.3536\; }} & {{- 0.3536} - {0.3536\; }} \end{matrix}$ $C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$

$C_{3,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ $C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {{0 + {0.5000\; }}\;} & {0 - {0.5000}} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$

$C_{7,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000\; }} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000\; }} & {0 - {0.5000\; }} & {0 + {0.5000\; }} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000\; }} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$ $C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000\; }} & 0.5000 \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000\; }} & {0 - {0.5000\; }} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000\; }} & {0 - {0.5000\; }} \end{matrix}$

$\mspace{20mu} {C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000\; }} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000\; }} \end{matrix}}$ $\mspace{20mu} {C_{12,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000\; }} \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & 0.5000 & {0 - {0.5000\; }} \end{matrix}}$ $C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ $\mspace{20mu} {C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}}$ $\mspace{20mu} {C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}}$

$C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}$ $C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 & {0 - {0.5000}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

For the third example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to the third example may be designed by combining subsets of two matrices W₃ and W₆ as follows:

$\begin{matrix} \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \\ 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

For example, where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the multi-user MIMO communication system according to the third example may be expressed using a transmission rank, as given by the following Table 8:

Transmit Codebook Index Rank 1 1 M₁=W3(:,1) 2 M₂=W3(:,2) 3 M₃=W3(:,3) 4 M₄=W3(:,4) 5 M₅=W6(:,1) 6 M₆=W6(:,2) 7 M₇=W6(:,3) 8 M₈=W6(:,4)

The codewords included in the above Table 8 may be expressed equivalently as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $M_{4} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station that includes four physical antennas.

Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 9:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1)=W1(;,2) 2 C_(2,1)=W1(;,3) 3 C_(3,1)=W1(;,4) 4 C_(4,1)=W2(;,2) 5 C_(5,1)=W2(;,3) 6 C_(6,1)=W2(;,4) 7 C_(7,1)=W3(;,1) 8 C_(8,1)=W4(;,1) 9 C_(9,1)=W5(;,1) 10 C_(10,1)=W5(;,2) 11 C_(11,1)=W5(;,3) 12 C_(12,1)=W5(;,4) 13 C_(13,1)=W6(;,1) 14 C_(14,1)=W6(;,2) 15 C_(15,1)=W6(;,3) 16 C_(16,1)=W6(;,4)

The codewords included in the above Table 9 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{3,1} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {03536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

A fourth example of a codebook used in a single user MIMO communication system or a multi-user MIMO communication system where a number of physical antennas of a base station is four, is illustrated below.

In this example, a rotation matrix U_(rot) and a QPSK DFT matrix may be defined as follows.

$U_{rot} = {\frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & {- 1} & 0 \\ 0 & 1 & 0 & {- 1} \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{bmatrix}}$ and ${DFT} = {0.5*{\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ 1 & {- j} & {- 1} & j \end{bmatrix}.}}$

Also, diag(a, b, c, d) is a 4×4 matrix, and diagonal elements of diag(a, b, c, d) are a, b, c, and d, and all the remaining elements are zero.

Codewords W₁, W₂, W₃, W₄, W₅, and W₆ may be defined as follows:

$\begin{matrix} \begin{matrix} {W_{1} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{2} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{4\;} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ j & {- j} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & {- 1} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ j & {- j} & 1 & {- 1} \\ 1 & 1 & {- 1} & {- 1} \\ j & {- j} & {- 1} & 1 \end{bmatrix}}} \end{matrix} & \; \\ {W_{5} = {{{{diag}\left( {1,1,1,{- 1}} \right)}*{DFT}}\mspace{34mu} = {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & j & {- 1} & {- j} \\ 1 & {- 1} & 1 & {- 1} \\ {- 1} & j & 1 & {- j} \end{bmatrix}}}} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

Where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the single user MIMO communication system according to the fourth example may be given by the following Table 10:

Transmit Transmission Transmission Transmission Transmission Codebook Index Rank 1 Rank 2 Rank 3 Rank 4 1 C_(1,1)=W1(;,2) C_(1,2)=W1(:,12) C_(1,3)=W1(;,1 2 3) C_(1,4)=W1(;,1 2 3 4) 2 C_(2,1)=W1(;,3) C_(2,2)=W1(:,13) C_(2,3)=W1(;,1 2 4) C_(2,4)=W2(;,1 2 3 4) 3 C_(3,1)=W1(;,4) C_(3,2)=W1(:,14) C_(3,3)=W1(;,1 3 4) C_(3,4)=W3(;,1 2 3 4) 4 C_(4,1)=W2(;,2) C_(4,2)=W1(:,23) C_(4,3)=W1(;,2 3 4) C_(4,4)=W4(;,1 2 3 4) 5 C_(5,1)=W2(;,3) C_(5,2)=W1(:,24) C_(5,3)=W2(;,1 2 3) C_(5,4)=W5(;,1 2 3 4) 6 C_(6,1)=W2(;,4) C_(6,2)=W1(:,34) C_(6,3)=W2(;,1 2 4) C_(6,4)=W6(;,1 2 3 4) 7 C_(7,1)=W3(;,1) C_(7,2)=W2(:,13) C_(7,3)=W2(;,1 3 4) n/a 8 C_(8,1)=W4(;,1) C_(8,2)=W2(:,14) C_(8,3)=W2(;,2 3 4) n/a 9 C_(9,1)=W5(;,1) C_(9,2)=W2(:,23) C_(9,3)=W3(;,1 2 3) n/a 10 C_(10,1)=W5(;,2) C_(10,2)=W2(:,24) C_(10,3)=W3(;,1 3 4) n/a 11 C_(11,1)=W5(;,3) C_(11,2)=W3(:,13) C_(11,3)=W4(;,1 2 3) n/a 12 C_(12,1)=W5(;,4) C_(12,2)=W3(:,14) C_(12,3)=W4(;,1 3 4) n/a 13 C_(13,1)=W6(;,1) C_(13,2)=W4(:,13) C_(13,3)=W5(;,1 2 3) n/a 14 C_(14,1)=W6(;,2) C_(14,2)=W4(:,14) C_(14,3)=W5(;,1 3 4) n/a 15 C_(15,1)=W6(;,3) C_(15,2)=W5(:,13) C_(15,3)=W6(;,1 2 4) n/a 16 C_(16,1)=W6(;,4) C_(16,2)=W6(:,24) C_(16,3)=W6(;,2 3 4) n/a

The codewords included in the above Table 10 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ {- 0.5000} \\ {- 0.5000} \end{matrix}$ $C_{3,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ {- 0.5000} \\ 0.5000 \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {0.3536 + {03536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {03536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {03536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {03536}} \end{matrix}$

$C_{1,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ $C_{4,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \end{matrix}$

$C_{5,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{6,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{7,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{8,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} \end{matrix}$

$C_{9,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}$ $C_{10,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{11,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 - {0.5000}} & 0.5000 \end{matrix}$ $C_{12,2} = \begin{matrix} 0.5000 & 0.5000 \\ {0 - {0.5000}} & 0.5000 \\ {- 0.5000} & 0.5000 \\ {0 + {0.5000}} & 0.5000 \end{matrix}$

$C_{13,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} \\ {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} \end{matrix}$ $C_{14,2} = \begin{matrix} 0.5000 & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} \\ {- 0.5000} & 0.5000 \\ 0.5000 & {0 + {0.5000}} \end{matrix}$ $C_{15,2} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ $C_{16,2} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

$C_{1,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ $C_{2,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{3,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ $C_{4,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}$ $C_{6,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{7,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{8,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$

$C_{9,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} \end{matrix}$ $C_{10,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{11,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} \end{matrix}$ $C_{12,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{13,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 \end{matrix}$ $C_{14,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000}} \end{matrix}$ $C_{15,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536} \end{matrix} \end{matrix}$ $C_{16,3} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536} \end{matrix} \end{matrix}$

$C_{1,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \end{matrix}$ $C_{2,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{3,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ $C_{4,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} & 0.5000 \end{matrix}$

$C_{5,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 & {0 - {0.5000}} \end{matrix}$ $C_{6,4} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536} \end{matrix} \end{matrix}$

For the fourth example, where the number of physical antennas of the base station is four, the codebook used in the multi-user MIMO communication system according to the fourth example may be designed by combining subsets of two matrices W₃ and W₆ as follows:

$\begin{matrix} \begin{matrix} {W_{3} = {\frac{1}{\sqrt{2}}*U_{rot}*\begin{bmatrix} 1 & 1 & 0 & 0 \\ 1 & {- 1} & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & j & {- j} \end{bmatrix}*\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & {- 1} & 0 \\ 0 & 0 & 0 & {- 1} \end{bmatrix}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & {- 1} & j & {- j} \\ 1 & 1 & {- 1} & {- 1} \\ 1 & {- 1} & {- j} & j \end{bmatrix}}} \end{matrix} & \; \\ \begin{matrix} {W_{6} = {{{diag}\left( {1,\frac{\left( {1 + j} \right)}{\sqrt{2}},j,\frac{\left( {{- 1} + j} \right)}{\sqrt{2}}} \right)}*{DFT}}} \\ {= {0.5*\begin{bmatrix} 1 & 1 & 1 & 1 \\ \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} \\ j & {- j} & j & {- j} \\ \frac{\left( {{- 1} + j} \right)}{\sqrt{2}} & \frac{\left( {1 + j} \right)}{\sqrt{2}} & \frac{\left( {1 - j} \right)}{\sqrt{2}} & \frac{\left( {{- 1} - j} \right)}{\sqrt{2}} \end{bmatrix}}} \end{matrix} & \; \end{matrix}$

Where the number of physical antennas of the base station is four, the matrices or the codewords included in the codebook used in the multi-user MIMO communication system according to the fourth example may be expressed using a transmission rank, as given by the following Table 11:

codeword used at Transmit Codebook the mobile station Index for quantization 1 M₁=W3(;,1) 2 M₂=W3(;,2) 3 M₃=W3(;,3) 4 M₄=W3(;,4) 5 M₅=W6(;,1) 6 M₆=W6(;,2) 7 M₇=W6(;,3) 8 M₈=W6(;,4)

The codewords included in the above Table 11 may be expressed equivalently as follows:

$M_{1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $M_{2} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $M_{3} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$ $M_{4} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$

$M_{5} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $M_{6} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $M_{7} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $M_{8} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

In this example, the codebook used in the multi-user MIMO communication system performing non-unitary precoding has a base station that includes four physical antennas.

Codewords included in the codebook used in the multi-user MIMO communication system performing non-unitary precoding may be given by the following Table 12:

Transmit Codebook Transmission Index Rank 1 1 C_(1,1)=W1(;,2) 2 C_(2,1)=W1(;,3) 3 C_(3,1)=W1(;,4) 4 C_(4,1)=W2(;,2) 5 C_(5,1)=W2(;,3) 6 C_(6,1)=W2(;,4) 7 C_(7,1)=W3(;,1) 8 C_(8,1)=W4(;,1) 9 C_(9,1)=W5(;,1) 10 C_(10,1)=W5(;,2) 11 C_(11,1)=W5(;,3) 12 C_(12,1)=W5(;,4) 13 C_(13,1)=W6(;,1) 14 C_(14,1)=W6(;,2) 15 C_(15,1)=W6(;,3) 16 C_(16,1)=W6(;,4)

The codewords included in the above Table 12 may be expressed equivalently as follows:

$C_{1,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{2,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ {- 0.5000} \\ {- 0.5000} \end{matrix}$ $C_{3,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ {- 0.5000} \\ 0.5000 \end{matrix}$ $C_{4,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

$C_{5,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$ $C_{6,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{7,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{8,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$

$C_{9,1} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ $C_{10,1} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \\ {0 + {0.5000}} \end{matrix}$ $C_{11,1} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ $C_{12,1} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \\ {0 - {0.5000}} \end{matrix}$

$C_{13,1} = \begin{matrix} 0.5000 \\ {03536 + {0.3536}} \\ {0 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ $C_{14,1} = \begin{matrix} 0.5000 \\ {{- 03536} + {0.3536}} \\ {0 - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ $C_{15,1} = \begin{matrix} 0.5000 \\ {{- 03536} - {0.3536}} \\ {0 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ $C_{16,1} = \begin{matrix} 0.5000 \\ {03536 - {0.3536}} \\ {0 - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

Various examples of codebooks and codewords according to a transmission rank and a number of antennas of a base station in a single user MIMO communication system and a multi-user MIMO communication system, have been described above. The aforementioned codewords may be modified using various types of schemes or shapes, and thus are not limited to the aforementioned examples. A similar codebook, or an equivalent codebook may be obtained by changing the phase of the columns of the codewords, for example, by multiplying the columns of the codewords by a complex exponential. As another example, one may multiply the columns of the codewords by ‘−1’.

In some embodiments, the performance or the properties of the codebook may not change when the phases of the columns of the codewords have been changed. Accordingly, it is understood that a codebook generated by changing the phases of the columns of the codewords may be the same as a codebook including the original codewords prior to the changing of the phases. Also, it is understood that a codebook generated by swapping columns of the original codewords of a codebook may be the same as the codebook including the original codewords prior to the swapping. Specific values that are included in the codebook generated by changing the phases of the columns of the original codewords, or in the codebook generated by swapping the columns of the original codewords, will be omitted herein for conciseness.

A fifth example of a codebook used in a single user MIMO communication system where the number of physical antennas of a base station is eight is shown in the following Equation 7:

$\begin{matrix} {W_{0} = {{\frac{1}{\sqrt{8}}\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & ^{j\; {\pi/4}} & ^{{j\pi}/2} & ^{j\; 3{\pi/4}} & ^{j\pi} & ^{{j5\pi}/4} & ^{{j3\pi}/2} & ^{j\; 7{\pi/4}} \\ 1 & ^{j\; {\pi/2}} & ^{j\pi} & ^{j\; 3{\pi/2}} & 1 & ^{j\; {\pi/2}} & ^{j\pi} & ^{j\; 3{\pi/2}} \\ 1 & ^{j\; 3{\pi/4}} & ^{{j3\pi}/2} & ^{j\; {\pi/4}} & ^{j\; \pi} & ^{j\; 7{\pi/4}} & ^{{j\pi}/2} & ^{{j5\pi}/4} \\ 1 & ^{j\; \pi} & 1 & ^{j\pi} & 1 & ^{j\; \pi} & 1 & ^{j\; \pi} \\ 1 & ^{{j5\pi}/4} & ^{j\; {\pi/2}} & ^{j\; 7{\pi/4}} & ^{j\pi} & ^{j\; {\pi/4}} & ^{j\; 3{\pi/2}} & ^{j\; 3{\pi/4}} \\ 1 & ^{{j6\pi}/4} & ^{j\pi} & ^{j\; {\pi/2}} & 1 & ^{j\; 3{\pi/2}} & ^{j\pi} & ^{j\; {\pi/2}} \\ 1 & ^{{j7\pi}/4} & ^{{j3\pi}/2} & ^{j\; 5{\pi/4}} & ^{j\; \pi} & ^{{j3\pi}/4} & ^{{j\pi}/2} & ^{j\; {\pi/4}} \end{bmatrix}}.}} & (7) \end{matrix}$

Matrices included in the codebook for the single user MIMO communication system may be determined using a transmission rank, as given by the following Table 13:

Transmit Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 C1,1 = W0(;,1) C1,2 = W0(;,1 2) C1,3 = W0(;,1 2 3) C1,4 = W0(;,1 2 3 4) 2 C2,1 = W0(;,2) C2,2 = W0(;,3 4) C2,3 = W0(;,3 4 5) C2,4 = W0(;,3 4 5 6) 3 C3,1 = W0(;,3) C3,2 = W0(;,5 6) C3,3 = W0(;,5 6 7) C3,4 = W0(;,5 6 7 8) 4 C4,1 = W0(;,4) C4,2 = W0(;,7 8) C4,3 = W0(;,7 8 1) C4,4 = W0(;,7 8 1 2) 5 C5,1 = W0(;,5) C5,2 = W0(;,1 3) C5,3 = W0(;,1 3 5) C5,4 = W0(;,1 3 5 7) 6 C6,1 = W0(;,6) C6,2 = W0(;,2 4) C6,3 = W0(;,2 4 6) C6,4 = W0(;,2 4 6 8) 7 C7,1 = W0(;,7) C7,2 = W0(;,5 7) C7,3 = W0(;,5 7 1) C7,4 = W0(;,5 7 1 4) 8 C8,1 = W0(;,8) C8,2 = W0(;,6 8) C8,3 = W0(;,6 8 2) C8,4 = W0(;,6 8 2 3)

Referring to the above Table 13, where the transmission rank is 4, the precoding matrix may be generated based on, for example, W₀(;,1 2 3 4), W₀(;,3 4 5 6), W₀(;,5 6 7 8), W₀(;,7 8 1 2), W₀(;,1 3 5 7), W₀(;,2 4 6 8), W₀(;,5 7 1 4), and W₀(;,6 8 2 3). In this example, W_(k)(;,n m o p) denotes a matrix that includes an n^(th) column vector, an m^(th) column vector, an o^(th) column vector, and a p^(th) column vector of W_(k).

Where the transmission rank is 3, the precoding matrix may be generated based on, for example, W₀(;,1 2 3), W₀(;,3 4 5), W₀(;,5 6 7), W₀(;,7 8 1), W₀(;,1 3 5), W₀(;,2 4 6), W_(o)(;,5 7 1), and W₀(;,6 8 2). In this example, W_(k)(;,n m o) denotes a matrix that includes the n^(th) column vector, the m^(th) column vector, and the o^(th) column vector of W_(k).

Where the transmission rank is 2, the precoding matrix may be generated based on, for example, W₀(;,1 2), W₀(;,3 4), W₀(;,5 6), W₀(;,7 8), W₀(;,1 3), W₀(;,2 4), W₀(;,5 7), and W₀(;,6 8). In this example, W_(k)(;,n m) denotes a matrix that includes the n^(th) column vector and the m^(th) column vector of W_(k).

Where the transmission rank is 1, the precoding matrix may be generated based on, for example, W₀(;,1), W₀(;,2), W₀(;,3), W₀(;,4), W₀(;,5), W₀(;,6), W₀(;,7), and W₀(;,8). In this example, W_(k)(;,n) denotes the n^(th) column vector of W_(k).

Design of a 6-Bit Codebook

According to an increase in a number of elements, for example, matrices or vectors, included in a codebook, a feedback overhead may increase whereas a quantization error may decrease. The above tables disclose 4-bit codebooks, each including a plurality of elements. Hereinafter, 6-bit codebooks, including a plurality of elements, for example, 64 elements, will be discussed.

1. First Scheme to Design a 6-Bit Codebook:

In this example, it is assumed that the number of physical antennas of a base station is four, a single user MIMO communication system uses the 4-bit codebooks disclosed in the above Table 1, and a multi-user MIMO communication system uses the codebooks disclosed in the above Table 2:

(1) Operation 1:

The following 4-bit codebook corresponding to the transmission rank 1 may be obtained from the 4-bit codebook disclosed in the above Table 1. In this example, the 4-bit codebook corresponding to the transmission rank 1, disclosed in the above Table 1, is referred to as a base codebook. The base codebook may be expressed using a language according to a numerical analysis program, for example, MATLAB®, manufactured by MathWorks Inc. of Natick, Mass. The numerical analysis language may be, for example, as follows:

rotation_matrix=1/sqrt(2)*[1,0,−1,0;0,1,0,−1;1,0,1,0;0,1,0,1]; W(:,:,1)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1,−1]; W(:,:,2)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,3)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,4)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1,−1]; DFT=1/sqrt(4)*[1,1,1,1;1,1i,−1,−1i;1,−1,1,−1;1,−1i,−1,1i]; W(:,:,5)=diag([1,1,1,−1])*DFT; W(:,:,6)=diag([1,(1+1i)/sqrt(2),1i,(−1+1i)/sqrt(2)])*DFT; base_cbk(:,1:4)=W(:,[1:4],1); base_cbk(:,5:8)=W(:,[1:4],2); base_cbk(:,9:12)=W(:,[1:4],5); base_cbk(:,13:16)=W(:,[1:4],6);

In this example, ‘rotation_matrix’ denotes a rotation matrix U_(rot), ‘sqrt(x)’ denotes √{square root over (x)}, and ‘DFT’ denotes a QPSK DFT matrix. ‘base_cbk(:,a:b)’ denotes an a^(th) element or vector, through a b^(th) element or vector of the base codebook. ‘W(:,[x:y],k)’ denotes a column x^(th) through a y^(th) element among elements of W_(k). For example, ‘base_cbk(:,1:4)’ denotes first, second, third, and fourth columns of the base codebook. ‘W(:,[1:4],1)’ denotes first, second, third, and fourth columns of W₁.

(2) Operation 2:

A local codebook local_cbk may be defined, for example, as follows:

local_cbk= [0.2778+0.4157i   0.2778+0.4157i   0.2778+0.4157i   0.2778+0.4157i −0.0975+0.4904i −0.4904−0.0975i   0.0975−0.4904i   0.4904+0.0975i −0.0975+0.4904i   0.0975−0.4904i −0.0975+0.4904i   0.0975−0.4904i   0.4904+0.0975i   0.0975−0.4904i −0.4904−0.0975i −0.0975+0.4904i];

(3) Operation 3:

The local codebook local_cbk may be scaled to obtain a localized codebook localized_cbk. An exemplary scaling process is as follows:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha= 0.5960

In this example, abs(local_cbk) denotes a magnitude of the local codebook local_cbk. The localized codebook localized_cbk may be expressed as follows:

localized_cbk= [sqrt(1-alpha{circumflex over ( )}2*(1-(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1- alpha{circumflex over ( )}2*(1-(r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1- alpha{circumflex over ( )}2*(1-(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1- alpha{circumflex over ( )}2*(1-(r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)* phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)* phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)* phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2) localized_cbk(:,k)=localized_cbk(:,k)/norm(localized_cbk(:,k)); end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

for k=1:16 [U,S,V]=svd(base_cbk(:,k)); %, where a singular value decomposition (SVD) is performed with respect to the elements of the base codebook. R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]. % , where R denotes a rotation matrix that rotates the normalized localized codebook localized_cbk around the base codebook. rotated_localized=R′*localized_cbk(:,1:3);%, where rotated_localized is obtained by rotating localized_cbk, and only first three vectors of localized_cbk are rotated. final_cbk(:,(k−1)*4+1:k*4)=[base_cbk(:,k),rotated_localized]; %, where the base codebook is maintained as a centroid and the base codebook is included in a final 6-bit codebook end;

The final 6-bit codebook final_cbk_opt may be generated as follows through the aforementioned operations 1 through 4. The final 6-bit codebook final_cbk_opt may include 64 column vectors.

final_cbk_opt = Columns  1  through  4 $\begin{matrix} 0.5000 & {0.3260 + {0.6774}} & {0.1499 + {0.0347}} & {0.0918 + {0.3270}} \\ 0.5000 & {0.3254 + {0.1709}} & {0.5009 + {0.3071}} & {0.1311 + {0.6387}} \\ 0.5000 & {0.3254 + {0.1709}} & {0.1505 + {0.5412}} & {0.2473 + {0.0541}} \\ 0.5000 & {{- 0.0250} + {0.4051}} & {0.1505 + {0.5412}} & {0.4815 + {0.4045}} \end{matrix}$ Columns  5  through  8 $\begin{matrix} 0.5000 & {0.0918 + {0.3270}} & {0.3841 + {0.3851}} & {0.3260 + {0.6774}} \\ {- 0.5000} & {{- 0.1311} - {0.6387}} & {0.0056 - {0.30761}} & {{- 0.3254} - {0.1709}} \\ 0.5000 & {0.2473 + {0.0541}} & {0.2285 + {0.6580}} & {0.3254 + {0.1709}} \\ {- 0.5000} & {{- 0.4815} - {0.4045}} & {{- 0.3448} - {0.0735}} & {0.0250 - {0.4051}} \end{matrix}$ Columns  9  through  12 $\begin{matrix} {- 0.5000} & {{- 0.0918} - {0.3270}} & {{- 0.0337} - {0.6193}} & {{- 0.4422} - {0.0928}} \\ {- 0.5000} & {{- 0.2473} - {0.0541}} & {{- 0.4621} - {0.5019}} & {{- 0.2479} - {0.5606}} \\ 0.5000 & {0.1311 + {0.6387}} & {0.2280 + {0.1515}} & {0.2479 + {0.5606}} \\ 0.5000 & {0.4815 + {04045}} & {0.2280 + {0.1515}} & {0.0137 + {0.2102}} \end{matrix}$ Columns  13  through  16 $\begin{matrix} {- 0.5000} & {{- 0.4422} - {0.0928}} & {{- 0.3841} - {0.3851}} & {{- 0.0918} - {0.3270}} \\ 0.5000 & {0.2479 + {0.5606}} & {{- 0.0056} + {0.3076}} & {0.2473 + {0.0541}} \\ 0.5000 & {0.2479 + {0.5606}} & {0.3448 + {0.0735}} & {0.1311 + {0.6387}} \\ {- 0.5000} & {{- 0.0137} - {0.2102}} & {{- 0.2285} - {0.6580}} & {{- 0.4815} - {0.4045}} \end{matrix}$

Columns  17  through  20 $\begin{matrix} 0.5000 & {0.3841 + {0.3851}} & {0.0918 + {0.3270}} & {0.337 + {0.6193}} \\ {0 + {0.5000}} & {{- 0.3076} - {0.0056}} & {{- 0.0541} + {0.2473}} & {{- 0.5019} + {0.4621}} \\ 0.5000 & {0.3448 + {0.0735}} & {0.1311 + {0.6387}} & {0.2280 + {0.1515}} \\ {0 + {0.5000}} & {{- 0.6580} + {0.2285}} & {{- 0.4045} + {0.4815}} & {{- 0.1515} + {0.2280}} \end{matrix}$ Columns  21  through  24 $\begin{matrix} 0.5000 & {0.0337 + {0.6193}} & {0.4422 + {0.0928}} & {0.3841 + {0.3851}} \\ {0 - {0.5000}} & {0.5019 - {0.4621}} & {0.5606 - {0.2479}} & {0.3076 + {0.0056}} \\ 0.5000 & {0.2280 + {0.1515}} & {0.2479 + {0.5606}} & {0.3448 + {0.0735}} \\ {0 - {0.5000}} & {0.1515 - {0.2280}} & {0.2102 - {0.0137}} & {0.6580 - {0.2285}} \end{matrix}$ Columns  25  through  28 $\begin{matrix} {- 0.5000} & {{- 0.3841} - {0.3851}} & {{- 0.3260} - {0.6774}} & {{- 0.1499} - {0.0347}} \\ {0 - {0.5000}} & {0.3076 + {0.0056}} & {0.1709 - {0.3254}} & {0.3071 - {0.5009}} \\ 0.5000 & {0.2285 + {0.6580}} & {0.3254 + {0.1709}} & {0.1505 + {0.5412}} \\ {0 + {0.5000}} & {{- 0.0735} + {0.3448}} & {{- 0.4051} - {0.0250}} & {{- 0.5412} + {0.1505}} \end{matrix}$ Columns  29  through  32 $\begin{matrix} {- 0.5000} & {{- 0.1499} - {0.0347}} & {{- 0.0918} - {0.3270}} & {{- 0.3841} - {0.3851}} \\ {0 + {0.5000}} & {{- 0.3071} + {0.5009}} & {{- 0.6387} + {0.1311}} & {{- 0.3076} - {0.0056}} \\ 0.5000 & {0.1505 + {0.5412}} & {0.2473 + {0.0541}} & {0.2285 + {0.6580}} \\ {0 - {0.5000}} & {0.5412 - {0.1505}} & {0.4045 - {0.4815}} & {0.0735 - {0.3448}} \end{matrix}$

Columns  33  through  36 $\begin{matrix} 0.5000 & {0.0337 + {0.6193}} & {0.0918 + {0.3270}} & {0.3841 + {0.3851}} \\ 0.5000 & {0.2280 + {0.1515}} & {0.4815 + {0.4045}} & {0.2285 + {0.6580}} \\ 0.5000 & {0.2280 + {0.1515}} & {0.1311 + {0.6387}} & {0.3448 + {0.0735}} \\ {- 0.5000} & {{- 0.4621} - {05019}} & {{- 0.2473} - {0.0541}} & {0.0056 - {0.3076}} \end{matrix}$ Columns  37  through  40 $\begin{matrix} 0.5000 & {0.4422 + {0.0928}} & {0.0337 + {0.6193}} & {0.0918 + {0.3270}} \\ {0 + {0.5000}} & {{- 0.2102} + {0.0137}} & {{- 0.1515} + {0.2280}} & {{- 0.4045} + {0.4815}} \\ {- 0.5000} & {{- 0.2479} - {0.5606}} & {{- 0.2280} - {0.1515}} & {{- 0.1311} - {06387}} \\ {0 + {0.5000}} & {{- 0.5606} + {0.2479}} & {{- 0.5019} + {0.4621}} & {{- 0.0541} + {0.2473}} \end{matrix}$ Columns  41  through  44 $\begin{matrix} 0.5000 & {0.3841 + {0.3851}} & {0.4422 + {0.0928}} & {0.0337 + {0.6193}} \\ {- 0.5000} & {{- 0.2285} - {0.6580}} & {{- 0.0137} - {0.2102}} & {{- 0.2280} - {01515}} \\ 0.5000 & {0.3448 + {0.0735}} & {0.2479 + {0.5606}} & {0.2280 + {0.1515}} \\ 0.5000 & {{- 0.0056} + {0.3076}} & {0.2479 + {0.5606}} & {0.4621 + {0.5019}} \end{matrix}$ Columns  45  through  48 $\begin{matrix} 0.5000 & {0.0918 + {0.3270}} & {0.3841 + {0.3851}} & {0.4422 + {0.0928}} \\ {0 - {0.5000}} & {0.4045 - {0.4815}} & {0.6580 - {0.2285}} & {0.2102 - {0.0137}} \\ {- 0.5000} & {{- 0.1311} - {0.6387}} & {{- 0.3448} - {0.0735}} & {{- 0.2479} - {0.5606}} \\ {0 - {0.5000}} & {0.0541 - {0.2473}} & {0.3076 + {0.0056}} & {0.5606 - {0.2479}} \end{matrix}$

Columns  49  through  52 $\begin{matrix} 0.5000 & {0.3841 + {0.3851}} & {{- 0.1560} + {0.4926}} & {0.3841 + {0.3851}} \\ {0.3536 + {0.3536}} & {0.0022 + {0.1690}} & {0.0837 + {0.4175}} & {{- 0.1140} + {0.7536}} \\ {0 + {0.5000}} & {{- 0.3076} - {0.0056}} & {{- 0.4597} + {0.3989}} & {{- 0.3076} - {0.0056}} \\ {{- 0.3536} + {0.3536}} & {{- 0.7536} - {0.1140}} & {{- 0.4175} + {0.0837}} & {{- 0.1690} + {0.0022}} \end{matrix}$ Columns  53  through  56 $\begin{matrix} 0.5000 & {0.3396 + {0.1614}} & {0.3841 + {0.3851}} & {{- 0.1560} + {0.4926}} \\ {{- 0.3536} + {0.3536}} & {{- 03400} - {0.3060}} & {{- 0.1690} + {0.0022}} & {{- 0.4175} + {0.0837}} \\ {0 - {0.5000}} & {0.3493 - {0.5641}} & {0.3076 + {0.0056}} & {0.4597 - {0.3989}} \\ {0.3536 + {0.3536}} & {{- 0.3060} + {0.3400}} & {{- 0.1140} + {0.7536}} & {0.0837 + {0.4175}} \end{matrix}$ Columns  57  through  60 $\begin{matrix} 0.5000 & {0.3841 + {0.3851}} & {0.3396 + {0.1614}} & {0.3841 + {0.3851}} \\ {{- 0.3536} - {0.3536}} & {0.1140 - {0.7536}} & {0.3060 - {0.3400}} & {{- 0.0022} - {0.1690}} \\ {0 + {0.5000}} & {{- 0.3076} - {0.0056}} & {{- 0.3493} + {0.5641}} & {{- 0.3076} - {0.0056}} \\ {0.3536 - {0.3536}} & {0.1690 - {0.0022}} & {0.3400 + {0.3060}} & {0.7536 + {0.1140}} \end{matrix}$ Columns  61  through  64 $\begin{matrix} 0.5000 & {{- 0.1560} + {0.4926}} & {03841 + {0.3851}} & {0.3396 + {0.1614}} \\ {0.3536 - {0.3536}} & {0.4175 - {0.0837}} & {0.7536 + {0.1140}} & {0.3400 + {0.3060}} \\ {0 - {0.5000}} & {0.4597 - {0.3989}} & {0.3076 + {0.0056}} & {0.3493 - {0.5641}} \\ {{- 0.3536} - {0.3536}} & {{- 0.0837} - {0.4175}} & {{- 0.0022} - {0.1690}} & {0.3060 - {0.3400}} \end{matrix}$

2. Second Scheme to Design a 6-Bit Codebook:

(1) Operation 1:

Like the first scheme to design the 6-bit codebook, a base codebook may be designed, for example, as follows:

rotation_matrix=1/sqrt(2)*[1,0,−1,0;0,1,0,−1;1,0,1,0;0,1,0,1]; W(:,:,1)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1,−1]; W(:,:,2)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,3)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1,−1,0,0;0,0,1,1;0,0,1i,−1i]; W(:,:,4)=1/sqrt(2)*rotation_matrix*[1,1,0,0;1i,−1i,0,0;0,0,1,1;0,0,1,−1]; DFT=1/sqrt(4)*[1,1,1,1;1,1i,−1,−1i;1,−1,1,−1;1,−1i,−1,1i]; W(:,:,5)=diag([1,1,1,−1])*DFT; W(:,:,6)=diag([1,(1+1i)/sqrt(2),1i,(−1+1i)/sqrt(2)])*DFT; base_cbk(:,1:4)=W(:,[1:4],1); base_cbk(:,5:8)=W(:,[1:4],2); base_cbk(:,9:12)=W(:,[1:4],5); base_cbk(:,13:16)=W(:,[1:4],6);

(2) Operation 2:

A local codebook local_cbk may be defined, for example, as follows:

${{local\_ cbk} = \begin{bmatrix} \begin{matrix} {0.2778 +} \\ {0.4157} \end{matrix} & \begin{matrix} {0.2778 +} \\ {0.4157} \end{matrix} & \begin{matrix} {0.2778 +} \\ {0.4157} \end{matrix} & \begin{matrix} {0.2778 +} \\ {0.4157} \end{matrix} \\ \begin{matrix} {0.0000 +} \\ {0.5000} \end{matrix} & \begin{matrix} {{- 0.5000} +} \\ {0.0000} \end{matrix} & \begin{matrix} {{- 0.0000} -} \\ {0.5000} \end{matrix} & \begin{matrix} {0.5000 -} \\ {0.0000} \end{matrix} \\ \begin{matrix} {{- 0.2778} +} \\ {0.4157} \end{matrix} & \begin{matrix} {0.2778 -} \\ {0.4157} \end{matrix} & \begin{matrix} {{- 0.2778} +} \\ {0.4157} \end{matrix} & \begin{matrix} {0.2778 -} \\ {0.4157} \end{matrix} \\ 0.5000 & {0 - {0.5000}} & {- 0.5000} & {0 + {0.5000}} \end{bmatrix}};$

(3) Operation 3:

The local codebook local_cbk may be scaled to obtain a localized codebook localized_cbk. An exemplary scaling process is as follows:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha=0.5716

The localized codebook localized_cbk may be expressed, for example, as follows:

localized_cbk= [sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)* phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)* phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)* phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2) localized_cbk(:,k)=localized_cbk(:,k)/norm(localized_cbk(:,k)); end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

  R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]. %, where R denotes a rotation matrix that rotates the normalized localized codebook localized_cbk around the base codebook.   rotated_localized=R'*localized_cbk(:,1:3);%, where rotated_localized is obtained by rotating localized_cbk, and only first three vectors of localized_cbk are rotated.   final_cbk(:,(k−1)*4+1:k*4)=[base_cbk(:,k),rotated_localized]; %, where the base codebook is maintained as a centroid and the base codebook is included in a final 6-bit codebook.     end;

The final 6-bit codebook final_cbk_opt may be generated as follows through the aforementioned operations 1 through 4. The final 6-bit codebook final_cbk_opt may include 64 column vectors.

final_cbk_opt = Columns  1  through  4 $\begin{matrix} 0.5000 & {0.3049 + {0.6229}} & {0.1779 + {0.0995}} & {0.0919 + {0.3371}} \\ 0.5000 & {0.2626 + {0.1626}} & {0.5060 + {0.2740}} & {0.1673 + {0.6390}} \\ 0.5000 & {0.4213 + {0.2108}} & {0.0614 + {0.5116}} & {0.3261 + {0.1156}} \\ 0.5000 & {{- 0.0232} + {0.4484}} & {0.2202 + {0.5598}} & {0.4531 + {0.3532}} \end{matrix}$ Columns  5  through  8 $\begin{matrix} 0.5000 & {0.0919 + {03371}} & {0.4637 + {0.3853}} & {0.3049 + {0.6229}} \\ {- 0.5000} & {{- 0.1673} - {06390}} & {{- 0.0297} - {03692}} & {{- 0.2626} - {0.1626}} \\ 0.5000 & {0.3261 + {0.1156}} & {0.1567 + {0.6069}} & {0.4213 + {0.2108}} \\ {- 0.5000} & {{- 0.4531} - {0.3532}} & {{- 0.3155} - {0.0834}} & {0.0232 - {0.4484}} \end{matrix}$ Columns  9  through  12 $\begin{matrix} {- 0.5000} & {{- 0.1779} - {0.3853}} & {{- 0.0191} - {0.6229}} & {{- 0.4637} - {0.0995}} \\ {- 0.5000} & {{- 0.2202} - {0.0834}} & {{- 0.4531} - {0.4484}} & {{- 0.3155} - {0.5598}} \\ 0.5000 & {0.0614 + {0.6069}} & {0.3261 + {0.2108}} & {0.1567 + {0.5116}} \\ 0.5000 & {0.5060 + {0.3692}} & {0.1673 + {0.1626}} & {0.0297 + {0.2740}} \end{matrix}$ Columns  13  through  16 $\begin{matrix} {- 0.5000} & {{- 0.4637} - {0.0995}} & {{- 0.3049} - {03371}} & {{- 0.1779} - {03853}} \\ 0.5000 & {0.3155 + {0.5598}} & {{- 0.0232} + {0.3532}} & {0.2202 + {0.0834}} \\ 0.5000 & {0.1567 + {0.5116}} & {0.4213 + {0.1156}} & {0.0614 + {0.6069}} \\ {- 0.5000} & {{- 0.0297} - {0.2740}} & {{- 0.2626} - {0.6390}} & {{- 0.5060} - {0.3692}} \end{matrix}$

Columns  17  through  20 $\begin{matrix} 0.5000 & {0.3049 + {0.3371}} & {0.1779 + {0.3853}} & {0.0191 + {06229}} \\ {0 + {0.5000}} & {{- 0.3532} - {0.0232}} & {{- 0.0834} + {0.2202}} & {{- 0.4484} + {0.4531}} \\ 0.5000 & {0.4213 + {0.1156}} & {0.0614 + {0.6069}} & {0.3261 + {0.2108}} \\ {0 + {0.5000}} & {{- 0.6390} + {0.2626}} & {{- 0.3692} + {0.5060}} & {{- 0.1626} + {0.1673}} \end{matrix}$ Columns  21  through  24 $\begin{matrix} 0.5000 & {0.0191 + {0.6229}} & {0.4637 + {0.0995}} & {0.3049 + {0.3371}} \\ {0 - {0.5000}} & {0.4484 - {0.4531}} & {0.5598 - {0.3155}} & {0.3532 + {0.0232}} \\ 0.5000 & {0.3261 + {0.2108}} & {0.1567 + {0.5116}} & {0.4213 + {0.1156}} \\ {0 - {0.5000}} & {0.1626 - {0.1673}} & {0.2740 - {0.0297}} & {0.6390 - {0.2626}} \end{matrix}$ Columns  25  through  28 $\begin{matrix} {- 0.5000} & {{- 0.4637} - {0.3853}} & {{- 0.3049} - {0.6229}} & {{- 0.1779} - {0.0995}} \\ {0 - {0.5000}} & {0.3692 - {0.0297}} & {0.1626 - {0.2626}} & {0.2740 - {0.5060}} \\ 0.5000 & {0.1567 + {0.6069}} & {0.4213 + {0.2108}} & {0.0614 + {0.5116}} \\ {0 + {0.5000}} & {{- 0.0834} + {0.3155}} & {{- 0.4484} - {0.0232}} & {{- 0.5598} + {0.2202}} \end{matrix}$ Columns  29  through  32 $\begin{matrix} {- 0.5000} & {{- 0.1779} - {0.0995}} & {{- 0.0191} - {0.3371}} & {{- 0.4637} - {0.3853}} \\ {0 + {0.5000}} & {{- 0.2740} + {0.5060}} & {{- 0.6390} + {0.1673}} & {{- 0.3692} + {0.0297}} \\ 0.5000 & {0.0614 + {0.5116}} & {0.3261 + {0.1156}} & {0.1567 + {0.6069}} \\ {0 - {0.5000}} & {0.5598 - {0.2202}} & {0.3532 - {0.4531}} & {0.0834 - {03155}} \end{matrix}$

Columns  33  through  36 $\begin{matrix} 0.5000 & {0.0191 + {0.6229}} & {0.1779 + {0.3853}} & {0.3049 + {0.3371}} \\ 0.5000 & {0.1673 + {0.1626}} & {0.5060 + {0.3692}} & {0.2626 + {0.6390}} \\ 0.5000 & {0.3261 + {0.2108}} & {0.0614 + {0.6069}} & {0.4213 + {0.1156}} \\ {- 0.5000} & {{- 0.4531} - {0.4484}} & {{- 0.2202} - {0.0834}} & {0.0232 - {0.3532}} \end{matrix}$ Columns  37  through  40 $\begin{matrix} 0.5000 & {0.4637 + {0.0995}} & {0.0191 + {0.6229}} & {0.1779 + {0.3853}} \\ {0 + {0.5000}} & {{- 0.2740} + {0.0297}} & {{- 0.1626} + {0.1673}} & {{- 0.3692} + {0.5060}} \\ {- 0.5000} & {{- 0.1567} - {0.5116}} & {{- 0.3261} - {0.2108}} & {{- 0.0614} - {0.6069}} \\ {0 + {0.5000}} & {{- 0.5598} + {0.3155}} & {{- 0.4484} + {0.4531}} & {{- 0.0834} + {0.2202}} \end{matrix}$ Columns  41  through  44 $\begin{matrix} 0.5000 & {0.3049 + {0.3371}} & {0.4637 + {0.0995}} & {0.0191 + {0.6229}} \\ {- 0.5000} & {{- 0.2626} - {0.6390}} & {{- 0.0297} - {0.2740}} & {{- 0.1673} - {0.1626}} \\ 0.5000 & {0.4213 + {0.1156}} & {0.1567 + {0.5116}} & {0.3261 + {0.2108}} \\ 0.5000 & {{- 0.0232} + {0.3532}} & {0.3155 + {0.5598}} & {0.4531 + {0.4484}} \end{matrix}$ Columns  45  through  48 $\begin{matrix} 0.5000 & {0.1779 + {0.3853}} & {0.3049 + {0.3371}} & {0.4637 + {0.0995}} \\ {0 - {0.5000}} & {0.3692 - {0.5060}} & {0.6390 - {0.2626}} & {0.2740 - {0.0297}} \\ {- 0.5000} & {{- 0.0614} - {0.6069}} & {{- 0.4213} - {0.1156}} & {{- 0.1567} - {0.5116}} \\ {0 - {0.5000}} & {0.0834 - {0.2202}} & {0.3532 + {0.0232}} & {0.5598 - {0.3155}} \end{matrix}$

Columns  49  through  52 $\begin{matrix} 0.5000 & {0.3602 + {0.4406}} & {{- 0.0795} + {0.4839}} & {0.3602 + {0.4406}} \\ {0.03536 + {0.3536}} & {{- 0.0754} + {0.1870}} & {0.0965 + {0.3794}} & {{- 0.0754} + {0.7586}} \\ {0 + {0.5000\iota}} & {{- 0.2289} + {0.0434}} & {{- 0.5609} + {0.3720}} & {{- 0.2289} + {0.0434}} \\ {{- 0.3536} + {0.3536}} & {{- 0.7586} - {0.0754}} & {{- 0.3794} + {0.0965}} & {{- 0.1870} - {0.0754}} \end{matrix}$ Columns  53  through  56 $\begin{matrix} 0.5000 & {0.3247 + {0.0797}} & {0.3602 + {0.4406}} & {{- 0.0795} + {0.4839}} \\ {{- 0.03536} + {0.3536}} & {{- 0.3794} - {0.2846}} & {{- 0.1870} - {0.0754}} & {{- 0.3794} + {0.0965}} \\ {0 - {0.5000}} & {0.4262 - {0.5068}} & {0.2289 - {0.0434}} & {0.5609 - {0.3720}} \\ {0.3536 + {0.3536}} & {{- 0.2846} + {0.3794}} & {{- 0.0754} + {0.7586}} & {0.0965 + {0.3794}} \end{matrix}$ Columns  57  through  60 $\begin{matrix} 0.5000 & {0.3602 + {0.4406}} & {0.3247 + {0.0797}} & {0.3602 + {0.4406}} \\ {{- 0.03536} - {0.3536}} & {0.0754 - {0.7586}} & {0.2846 - {0.3794}} & {0.0754 - {0.1870}} \\ {0 + {0.5000}} & {{- 0.2289} + {0.0434}} & {{- 0.4262} + {0.5068}} & {{- 0.2289} + {0.0434}} \\ {0.3536 - {0.3536}} & {0.1870 + {0.0754}} & {0.3794 + {0.2846}} & {0.7586 + {0.0754}} \end{matrix}$ Columns  61  through  64 $\begin{matrix} 0.5000 & {{- 0.0795} + {0.4839}} & {0.3602 + {0.4406}} & {0.3247 + {0.0797}} \\ {0.03536 - {0.3536}} & {0.3794 - {0.0965}} & {0.7586 + {0.0754}} & {0.3794 + {0.2846}} \\ {0 - {0.5000}} & {0.5609 - {0.3720}} & {0.2289 - {0.0434}} & {0.4262 - {0.5068}} \\ {{- 0.3536} - {0.3536}} & {{- 0.0965} - {0.3794}} & {0.0754 - {0.1870}} & {0.2846 - {0.3794}} \end{matrix}$

3. Third Scheme to Design a 6-Bit Codebook:

(1) Operation 1:

The following 4-bit codebook corresponding to the transmission rank 1 may be obtained from the 4-bit codebook disclosed in the above Table 1. In this example, the 4-bit codebook corresponding to the transmission rank 1, disclosed in the above Table 1, is referred to as a base codebook:

base_cbk(:,1:16)=[C _(1,1) . . . C _(16,1)] with C _(i,1) taken from table 1

(2) Operation 2:

Eight vectors included in two DFT matrices may be added to the base codebook of operation 1, so that the base codebook may include 24 vectors. An exemplary process for adding the vectors to the base codebook is as follows:

Nmat=4;% number of DFT matrices     indexmat = repmat(0:3,4,1);     DFT_mat = exp(j*2*pi*indexmat.*(indexmat′)/4)/sqrt(4);     for counter = 1:Nmat       offsnow = exp(j*2*pi*(0:(3))*(counter−1)/(4*Nmat));       Rnow = diag(offsnow);   Wbis(1:4,1:4,counter) = Rnow*DFT_mat;  end;  % rank 1  % 8 more precoders  base_cbk(:,17:20)= Wbis(:,[1:4],2); base_cbk(:,21:24)= Wbis(:,[1:4],4);

As shown in ‘base_cbk(:,17:20)=Wbis(:,[1:4],2);’ and ‘base_cbk(:,21:24)=Wbis(:,[1:4],4);’, 24 vectors may be included in the base codebook.

A local codebook local_cbk may be defined, for example, as follows:

${{local\_ cbk} = \begin{bmatrix} \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975} \end{matrix} & \begin{matrix} {{- 0.0975} -} \\ {0.4904} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975} \end{matrix} & \begin{matrix} {{- 0.4904} +} \\ {0.0975} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975} \end{matrix} \\ \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {{- 0.3536}} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} \end{bmatrix}};$

(3) Operation 3:

The local codebook local_cbk may be scaled to obtain the localized codebook localized_cbk. An exemplary scaling process is as follows:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha=0.9835;

Here, the localized codebook localized_cbk may be expressed, for example, as follows:

localized_cbk= [sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1− alpha{circumflex over ( )}2*(1−(r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)* phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)* phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)* phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2)     localized_cbk(:,k)=localized_cbk(:,k)/     norm(localized_cbk(:,k));   end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

for k=1:24       [U,S,V]=svd(base_cbk(:,k));       R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]       if k<=16         rotated_localized=R'*localized_cbk(:,1:2);%, where only first two vectors of localized_cbk are rotated and the base codebook is maintained as a centroid.    final_cbk_rank{1}(:,1,(k−1)*3+1:k*3)=    [base_cbk(:,k),rotated_localized];       else         rotated_localized=R'*localized_cbk(:,1);%, where only first two vectors of localized_cbk are rotated, and the base codebook is maintained as a centroid.         final_cbk_rank{1}(:,1,16*3+(k−17)*2+1:16*3+(k− 16)*2)=[base_cbk(:,k),rotated_localized];       end     end;

6 bits of final rank 1 codebook may be given, for example, by final_cbk_rank{1}.

A final rank 2 codebook, a final rank 3 codebook, and/or a final rank 4 codebook may also be obtained based on the final rank 1 codebook. For example, unitary matrices including columns of the final rank 1 codebook, different from the first 16 vectors of the base codebook, may be obtained. Thus, a total of 48 unitary matrices may be obtained. A total of 54 matrices may be obtained if W₁ through W₆, are added to the 48 unitary matrices. The 54 matrices may become elements of the final rank 4 codebook.

The final rank 2 codebook may be obtained by taking the first two columns from the 48 unitary matrices, and taking 16 matrices or codewords from the 4-bit rank 2 codebook disclosed in the above Table 1.

The final rank 3 codebook may be obtained by taking the second through the fourth columns from the 48 unitary matrices, and taking 16 matrices or codewords from the 4-bit rank 3 codebook disclosed in the above Table 1.

6 bits of the final rank 1 codebook, the final rank 2 codebook, the final rank 3 codebook, and the final rank 4 codebook may be expressed, for example, as follows:

Final Rank 1 Codebook:

${V\; 1\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},2} \right)} = \begin{matrix} {{- 0.2573} + {0.5267}} \\ {0.4306 + {0.2510}} \\ {0.4306 + {0.2510}} \\ {{- 0.3995} - {0.0009}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4182 - {0.2060}} \\ {0.2693 + {0.5849}} \\ {{- 0.3088} + {0.1985}} \\ {{- 0.1743} + {0.4504}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},5} \right)} = \begin{matrix} {{- 0.1228} + {0.0831}} \\ {0.4892 - {0.2949}} \\ {0.4754 + {0.1031}} \\ {{- 0.3409} - {0.5467}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},6} \right)} = \begin{matrix} {0.1663 + {0.6240}} \\ {0.0064 + {0.1030}} \\ {{- 0.3928} + {0.4752}} \\ {{- 0.4372} - {0.0316}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.1228 - {0.0831}} \\ {{- 0.4754} - {0.1031}} \\ {{- 0.4892} + {0.2949}} \\ {0.3409 + {0.5467}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4119 - {0.2376}} \\ {0.0073 - {0.7328}} \\ {0.3791 + {0.1546}} \\ {0.2445 - {0.0972}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.7073} - {0.3349}} \\ {{- 0.2125} + {0.3788}} \\ {{- 0.2125} + {0.3788}} \\ {0.0779 + {0.0648}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.0318} - {0.3722}} \\ {{- 0.0512} - {0.1869}} \\ {0.5269 + {0.1995}} \\ {0.3031 - {0.6431}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},13} \right)} = \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0318 + {0.3722}} \\ {0.1869 - {0.0512}} \\ {0.5269 + {0.1995}} \\ {{- 0.6431} - {0.3031}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {{- 0.1228} + {0.0831}} \\ {{- 0.1031} + {0.4754}} \\ {{- 0.4892} + {0.2949}} \\ {{- 0.5467} + {0.3409}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},16} \right)} = \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.4119} + {0.2376}} \\ {0.7328 + {0.0073}} \\ {0.3791 + {0.1546}} \\ {{- 0.0972} - {0.2445}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.7073 + {0.3349}} \\ {0.3788 + {0.2125}} \\ {{- 0.2125} + {0.3788}} \\ {{- 0.0648} + {0.0779}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},19} \right)} = {{\begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \\ {0 + {0.5000}} \end{matrix}V\; 1\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {{- 0.1663} - {0.6240}} \\ {{- 0.1030} + {0.0064}} \\ {{- 0.3928} + {0.4752}} \\ {{- 0.0316} + {0.4372}} \end{matrix}}$

${V\; 1\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.2473 - {0.5267}} \\ {0.2510 - {04306}} \\ {0.4306 + {0.2510}} \\ {0.0009 - {0.3995}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},22} \right)} = \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \\ {0 - {0.5000}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {{- 0.4182} + {0.2060}} \\ {{- 0.5849} + {0.2693}} \\ {{- 0.3088} + {0.1985}} \\ {0.4504 + {0.1743}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.1228 - {0.0831}} \\ {{- 0.2949} - {0.4892}} \\ {0.4754 + {0.1031}} \\ {0.5467 - {0.3409}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},25} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ 0.5000 \end{matrix} \\ 0.5000 \end{matrix} \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.6051} + {0.1790}} \\ {0.3147 + {0.1351}} \\ {0.3147 + {0.1351}} \\ {{- 0.1801} - {0.5787}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},27} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.0704 + {0.1417}} \\ {0.1534 + {0.7008}} \end{matrix} \\ {{- 0.4248} + {0.3144}} \end{matrix} \\ {{- 0.4053} + {0.1292}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},28} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0 + {0.5000}} \end{matrix} \\ {- 0.5000} \end{matrix} \\ {0 + {0.5000}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},29} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.5141 + {0.2763}} \\ {0.2189 + {0.1095}} \end{matrix} \\ {0.2769 - {0.3593}} \end{matrix} \\ {{- 0.6111} - {0.1423}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},30} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.6051} + {0.1790}} \\ {{- 0.1351} + {0.3147}} \end{matrix} \\ {{- 0.3147} - {0.1351}} \end{matrix} \\ {{- 0.5787} + {0.1801}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},31} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {- 0.5000} \end{matrix} \\ 0.5000 \end{matrix} \\ 0.5000 \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},32} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2250 + {0.4308}} \\ {0.3732 - {0.4108}} \end{matrix} \\ {0.5913 + {0.2190}} \end{matrix} \\ {{- 0.2387} - {0.0328}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},33} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.5141 + {0.2763}} \\ {{- 0.1095} + {0.2189}} \end{matrix} \\ {{- 0.2769} + {0.3593}} \end{matrix} \\ {{- 0.1423} + {0.6111}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},34} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0 - {0.5000}} \end{matrix} \\ {- 0.5000} \end{matrix} \\ {0 - {0.5000}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},35} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.0704 + {0.1417}} \\ {0.7008 - {0.1534}} \end{matrix} \\ {0.4248 - {0.3144}} \end{matrix} \\ {{- 0.1292} - {0.4053}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},36} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2250 + {0.4308}} \\ {0.4108 + {0.3732}} \end{matrix} \\ {{- 0.5913} - {0.2190}} \end{matrix} \\ {{- 0.0328} + {0.2387}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},37} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \end{matrix} \\ {0 + {0.5000}} \end{matrix} \\ {{- 0.3536} + {03536}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},38} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0376} + {0.4566}} \\ {0.2688 + {0.1481}} \end{matrix} \\ {0.1588 - {0.0744}} \end{matrix} \\ {{- 0.5917} - {0.5613}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},39} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4472} - {0.1208}} \\ {{- 0.0781} + {0.4936}} \end{matrix} \\ {{- 0.6133} - {0.0191}} \end{matrix} \\ {{- 0.3591} + {0.1738}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},40} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \end{matrix} \\ {0 - {0.5000}} \end{matrix} \\ {0.3536 + {0.3536}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},41} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.4535 + {0.1524}} \\ {0.1943 - {0.1220}} \end{matrix} \\ {0.7044 - {0.2811}} \end{matrix} \\ {{- 0.3738} - {0.0597}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},42} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0376} + {0.4566}} \\ {{- 0.1481} + {0.2688}} \end{matrix} \\ {{- 0.1588} + {0.0744}} \end{matrix} \\ {{- 0.5613} + {0.5917}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},43} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \end{matrix} \\ {0 + {0.5000}} \end{matrix} \\ {0.3536 - {0.3536}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},44} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2356 + {0.5396}} \\ {0.6510 - {0.4238}} \end{matrix} \\ {0.1311 + {0.0167}} \end{matrix} \\ {{- 0.0198} - {0.1791}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},45} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.4535 + {0.1524}} \\ {0.1220 + {0.1943}} \end{matrix} \\ {{- 0.7044} + {0.2811}} \end{matrix} \\ {{- 0.0597} + {0.3738}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},46} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \end{matrix} \\ {0 - {0.5000}} \end{matrix} \\ {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},47} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4472} - {0.1208}} \\ {0.4936 + {0.0781}} \end{matrix} \\ {0.6133 + {0.0191}} \end{matrix} \\ {{- 0.1738} - {0.3591}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},48} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.2356 + {0.5396}} \\ {0.4238 + {0.6510}} \end{matrix} \\ {{- 0.1311} - {0.0167}} \end{matrix} \\ {{- 0.1791} + {0.0198}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},49} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.4619 + {0.1913}} \end{matrix} \\ {0.3536 + {0.3536}} \end{matrix} \\ {0.1913 + {0.4619}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},50} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0628} + {0.5038}} \\ {0.3646 + {0.2226}} \end{matrix} \\ {0.2517 + {0.1533}} \end{matrix} \\ {{- 0.6562} - {0.2058}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},51} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.1913} + {0.4619}} \end{matrix} \\ {{- 0.3536} - {0.3536}} \end{matrix} \\ {0.4619 - {0.1913}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},52} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.4184 + {0.4842}} \\ {0.1085 - {0.0629}} \end{matrix} \\ {0.5948 - {0.4538}} \end{matrix} \\ {{- 0.0601} - {0.1068}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},53} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.4619} - {0.1913}} \end{matrix} \\ {0.3536 + {0.3536}} \end{matrix} \\ {{- 0.1913} - {0.4619}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},54} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.1083} + {0.4189}} \\ {0.6032 - {0.3824}} \end{matrix} \\ {0.2610 + {0.1225}} \end{matrix} \\ {{- 0.0597} - {0.4649}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},55} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.1913 - {0.4619}} \end{matrix} \\ {{- 0.3536} - {0.3536}} \end{matrix} \\ {{- 0.4619} + {0.1913}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},56} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0430} - {0.3791}} \\ {0.5314 - {0.0969}} \end{matrix} \\ {0.5001 - {0.1416}} \end{matrix} \\ {{- 0.3832} - {0.3817}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},57} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.1913 + {0.4619`}} \end{matrix} \\ {{- 0.3536} + {0.3536}} \end{matrix} \\ {{- 0.4619} - {0.1913}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},58} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.0197} + {0.5406}} \\ {0.1681 + {0.0640}} \end{matrix} \\ {0.2143 - {0.3250}} \end{matrix} \\ {{- 0.2387} - {0.6830}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},59} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.4619} + {0.1913}} \end{matrix} \\ {0.3536 - {0.3536}} \end{matrix} \\ {{- 0.1913} + {0.4619}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},60} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.1819 - {0.0589}} \\ {0.3368 - {0.1998}} \end{matrix} \\ {0.6565 - {0.0556}} \end{matrix} \\ {{- 0.5241} - {0.3183}} \end{matrix}$

${V\; 1\left( {\text{:},\text{:},61} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.1913} - {0.4619}} \end{matrix} \\ {{- 0.3536} + {0.3536}} \end{matrix} \\ {0.4619 + {0.1913}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},62} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {0.5307 + {0.2465}} \\ {0.6357 - {0.3878}} \end{matrix} \\ {0.1539 - {0.1260}} \end{matrix} \\ {{- 0.2499} - {0.0328}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},63} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} 0.5000 \\ {0.4619 - {0.1913}} \end{matrix} \\ {0.3536 - {0.3536}} \end{matrix} \\ {0.1913 - {0.4619}} \end{matrix}$ ${V\; 1\left( {\text{:},\text{:},64} \right)} = \begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4886} + {0.2995}} \\ {0.4671 + {0.2039}} \end{matrix} \\ {0.5829 + {0.1869}} \end{matrix} \\ {{- 0.1465} - {0.1250}} \end{matrix}$

Final Rank 2 Codebook:

${V\; 2\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},2} \right)} = {\begin{matrix} \begin{matrix} \begin{matrix} {{- 0.2573} + {0.5267}} \\ {0.4306 + {0.2510}} \end{matrix} \\ {0.4306 + {0.2510}} \end{matrix} \\ {{- 0.3995} - {0.0009}} \end{matrix}\begin{matrix} \begin{matrix} \begin{matrix} {{- 0.4516} - {0.0164}} \\ {{- 0.3631} - {0.4030}} \end{matrix} \\ {0.4237 + {0.2733}} \end{matrix} \\ {0.2522 + {0.4286}} \end{matrix}}$ ${V\; 2\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4182 - {0.2060}} & {0.7071 - {0.3537}} \\ {0.2693 + {0.5849}} & {{- 0.1549} - {0.4387}} \\ {{- 0.3088} + {0.1985}} & {0.3107 - {0.1994}} \\ {{- 0.1743} + {0.4504}} & {{- 0.1065} + {0.1039}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$

${V\; 2\left( {\text{:},\text{:},5} \right)} = \begin{matrix} {{- 0.1228} + {0.0831}} & {0.1911 + {0.1844}} \\ {0.4892 - {0.2949}} & {{- 0.6076} - {0.2220}} \\ {0.4754 + {0.1031}} & {0.0935 - {0.1294}} \\ {{- 0.3409} - {0.5467}} & {0.3520 - {0.6014}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},6} \right)} = \begin{matrix} {0.1663 + {0.6240}} & {{- 0.4390} - {0.0177}} \\ {0.0064 + {0.1030}} & {{- 0.2573} - {0.1623}} \\ {{- 0.3928} + {0.4752}} & {0.4550 + {0.6115}} \\ {{- 0.4372} - {0.0316}} & {0.0477 - {0.3623}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.1228 - {0.0831}} & {{- 0.8432} - {0.1719}} \\ {{- 0.4754} - {0.1031}} & {0.0004 - {0.0643}} \\ {{- 0.4892} + {0.2949}} & {0.0542 + {0.3715}} \\ {0.3409 + {0.5467}} & {{- 0.2612} + {0.2147}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4119 - {0.2376}} & {0.2307 - {0.3523}} \\ {0.0073 - {0.7328}} & {{- 0.1995} - {0.0992}} \\ {0.3791 + {0.1546}} & {{- 0.6261} + {0.4891}} \\ {0.2445 - {0.0972}} & {{- 0.3749} - {0.0349}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.7073} - {0.3349}} & {{- 0.3380} - {0.1860}} \\ {{- 0.2125} + {0.3788}} & {{- 0.2536} - {0.2825}} \\ {{- 0.2125} + {0.3788}} & {0.6905 - {0.2237}} \\ {0.0779 + {0.0648}} & {{- 0.3771} + {0.1950}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.0318} - {0.3722}} & {0.5870 + {0.5837}} \\ {{- 0.0512} - {0.1869}} & {0.1353 - {0.0302}} \\ {0.5269 + {0.1995}} & {0.1672 - {0.4033}} \\ {0.3031 - {0.6431}} & {0.1619 - {0.2805`}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},13} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0318 + {0.3722}} & {0.2910 + {0.0846}} \\ {0.1869 - {0.0512}} & {{- 0.5838} - {0.6826}} \\ {0.5269 + {0.1995}} & {{- 0.1084} + {0.1034}} \\ {{- 0.6431} - {0.3031}} & {0.0232 - {0.2799}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {{- 0.1228} + {0.0831}} & {0.4864 + {0.1062}} \\ {{- 0.1031} + {0.4754}} & {{- 0.3579} + {0.0709}} \\ {{- 0.4892} + {0.2949}} & {0.3743 - {0.4217}} \\ {{- 0.5467} + {0.3409}} & {{- 0.2104} + {0.5067}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.4119} + {0.2376}} & {{- 0.3475} + {0.0349}} \\ {0.7328 + {0.0073}} & {{- 0.3183} + {0.2743}} \\ {0.3791 + {0.1546}} & {0.3910 - {0.0682}} \\ {{- 0.0972} - {0.2445}} & {{- 0.5729} + {0.4644}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.7073 + {0.3349}} & {0.3580 + {0.4230}} \\ {0.3788 + {0.2125}} & {{- 0.5016} - {0.3133}} \\ {{- 0.2125} + {0.3788}} & {0.4516 - {0.0120}} \\ {{- 0.0648} + {0.0779}} & {0.2277 - {0.2953}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {{- 0.1663} - {0.6240}} & {{- 0.0285} - {0.3000}} \\ {{- 0.1030} + {0.0064}} & {{- 0.0146} + {0.4747}} \\ {{- 0.3928} + {0.4752}} & {0.1862 + {0.2754}} \\ {{- 0.0316} + {0.4372}} & {{- 0.4436} - {0.6134}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.2573 - {0.5267}} & {0.4288 + {0.0624}} \\ {0.2510 - {0.4306}} & {{- 0.2155} - {0.5900}} \\ {0.4306 + {0.2510}} & {0.0071 - {0.1770}} \\ {0.0009 - {0.3995}} & {0.1913 + {0.5913}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {{- 0.4182} + {0.2060}} & {{- 0.0368} - {0.04444}} \\ {{- 0.5849} + {0.2693}} & {0.0931 + {0.1023}} \\ {{- 0.3088} + {0.1985}} & {0.6755 + {0.2582}} \\ {0.4504 + {0.1743}} & {0.1401 + {0.6595}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.1228 - {0.0831}} & {0.5267 + {0.1302}} \\ {{- 0.2949} - {0.4892}} & {{- 0.3082} + {0.1349}} \\ {0.4754 + {0.1031}} & {{- 0.5798} - {0.2378}} \\ {0.5467 - {0.3409}} & {0.4428 + {0.0607}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},25} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.6051} + {0.1790}} & {{- 0.2867} + {0.4568}} \\ {0.3147 + {0.1351}} & {{- 0.3377} + {0.4028}} \\ {0.3147 + {0.1351}} & {{- 0.4367} - {0.4408}} \\ {{- 0.1801} - {0.5787}} & {0.2118 - {0.0548}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.0704 + {0.1417}} & {0.4489 - {0.4294}} \\ {0.1534 + {0.7008}} & {0.0676 - {0.2514}} \\ {{- 0.4248} + {0.3144}} & {{- 0.0562} - {0.3389}} \\ {{- 0.4053} + {0.1292}} & {{- 0.6352} + {0.1576}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},28} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.5141 + {0.2763}} & {0.0224 - {0.0857}} \\ {0.2189 + {0.1095}} & {0.4370 + {0.0749}} \\ {0.2769 - {0.3593}} & {0.0927 - {0.6134}} \\ {{- 0.6111} - {0.1423}} & {0.6031 - {0.2167}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {{- 0.6051} + {0.1790}} & {{- 0.2021} + {0.2515}} \\ {{- 0.1351} + {0.3147}} & {{- 0.2703} - {0.8413}} \\ {{- 0.3147} - {0.1351}} & {{- 0.3054} - {0.0715}} \\ {{- 0.5787} + {0.1801}} & {0.1019 + {0.0787}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},31} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.2250 + {0.4308}} & {{- 0.0056} + {0.0131}} \\ {0.3732 - {0.4108}} & {0.0211 + {0.5772}} \\ {0.5913 + {0.2190}} & {0.6155 - {0.2919}} \\ {{- 0.2387} - {0.0328}} & {0.3534 - {0.2780}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5141 + {0.2763}} & {0.0913 - {0.3481}} \\ {{- 0.1095} + {0.2189}} & {{- 0.5047} + {0.4712}} \\ {{- 0.2769} + {0.3593}} & {0.0947 - {0.5855}} \\ {{- 0.1423} + {0.6111}} & {{- 0.0727} + {0.1915}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},34} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.0704 + {0.1417}} & {0.2602 - {0.3335}} \\ {0.7008 - {0.1534}} & {0.3267 - {0.3041}} \\ {0.4248 - {0.3144}} & {0.1781 + {0.6675}} \\ {{- 0.1292} - {0.4053}} & {{- 0.1799} + {0.3348}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.2250 + {0.4308}} & {{- 0.1449} - {0.3470}} \\ {0.4108 + {0.3732}} & {{- 0.0692} + {0.7493}} \\ {{- 0.5913} - {0.2190}} & {{- 0.1504} + {0.4516}} \\ {{- 0.0328} + {0.2387}} & {0.1073 - {0.2330}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},37} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {{- 0.0376} + {0.4566}} & {{- 0.6780} - {0.3635}} \\ {0.2688 + {0.1481}} & {0.1160 + {0.0644}} \\ {0.1588 - {0.0744}} & {{- 0.3301} + {0.2541}} \\ {{- 0.5917} - {0.5613}} & {{- 0.4384} + {0.1574}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {{- 0.4472} - {0.1208}} & {0.2383 + {0.0963}} \\ {{- 0.0781} + {0.4936}} & {0.2810 + {0.0700}} \\ {{- 0.6133} - {0.0191}} & {0.2857 - {0.4528}} \\ {{- 0.3591} + {0.1738}} & {{- 0.4779} + {0.5788}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},40} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {- 0.5000} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4535 + {0.1524}} & {0.6361 - {0.0379}} \\ {0.1943 - {0.1220}} & {{- 0.3164} + {0.0577}} \\ {0.7044 - {0.2811}} & {0.0562 + {0.0352}} \\ {{- 0.3738} - {0.0597}} & {0.6782 - {0.1616}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {{- 0.0376} + {0.4566}} & {{- 0.6004} - {0.0993}} \\ {{- 0.1481} + {0.2688}} & {{- 0.4515} - {0.2985}} \\ {{- 0.1588} + {0.0744}} & {{- 0.1174} + {0.2648}} \\ {{- 0.5613} + {0.5917}} & {0.3667 + {0.3441}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},43} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2356 + {0.5396}} & {{- 0.2530} + {0.3209}} \\ {0.6510 - {0.4238}} & {{- 0.2885} - {0.0485}} \\ {0.1311 + {0.0167}} & {{- 0.0415} - {0.7379}} \\ {{- 0.0198} - {0.1791}} & {0.1649 - {0.4171}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.4535 + {0.1524}} & {0.2821 - {0.1279}} \\ {0.1220 + {0.1943}} & {{- 0.0004} - {0.0348}} \\ {{- 0.7044} + {0.2811}} & {{- 0.2923} + {0.0831}} \\ {{- 0.0597} + {0.3738}} & {{- 0.0749} - {0.8971}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},46} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {{- 0.4472} - {0.1208}} & {{- 0.4748} + {0.1205}} \\ {0.4936 + {0.0781}} & {0.2258 + {0.2637}} \\ {0.6133 + {0.0191}} & {{- 0.2650} + {0.3289}} \\ {{- 0.1738} - {0.3591}} & {{- 0.2798} + {0.6188}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {0.2356 + {0.5396}} & {{- 0.0180} - {0.6214}} \\ {0.4238 + {0.6510}} & {{- 0.0262} + {0.4269}} \\ {{- 0.1311} - {0.0167}} & {0.1087 + {0.3811}} \\ {{- 0.1791} + {0.0198}} & {{- 0.5069} + {0.1291}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},49} \right)} = \begin{matrix} 0.5000 & {0.0906 - {0.2217}} \\ {0.4619 + {0.1913}} & {{- 0.3471} + {0.3239}} \\ {0.3536 + {0.3536}} & {{- 0.2568} + {0.4202}} \\ {0.1913 + {0.4619}} & {0.6329 - {0.2724}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {{- 0.0628} + {0.5038}} & {{- 0.2542} - {0.0969}} \\ {0.3646 + {0.2226}} & {0.1965 + {0.0675}} \\ {0.2517 + {0.1533}} & {0.8534 + {0.0091}} \\ {{- 0.6562} - {0.2058}} & {0.3805 + {0.0988}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},51} \right)} = \begin{matrix} 0.5000 & {{- 0.0310} + {0.3127}} \\ {{- 0.1913} + {0.4619}} & {{- 0.4944} - {0.1906}} \\ {{- 0.3536} - {0.3536}} & {0.0394 + {0.1700}} \\ {0.4619 - {0.1913}} & {{- 0.1337} - {0.7565}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.4184 + {0.4842}} & {0.0769 + {0.4899}} \\ {0.1085 - {0.0629}} & {0.0623 - {0.0897}} \\ {0.5948 - {0.4538}} & {{- 0.4402} + {0.1739}} \\ {{- 0.0601} - {0.1068}} & {{- 0.7025} - {0.1568}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},53} \right)} = \begin{matrix} 0.5000 & {{- 0.1844} + {0.1638}} \\ {{- 0.4619} - {0.1913}} & {0.3389 - {0.3843}} \\ {0.3536 + {0.3536}} & {0.4409 - {0.5467}} \\ {{- 0.1913} - {0.4619}} & {{- 0.1168} - {0.4120}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {{- 0.1083} + {0.4189}} & {{- 0.1461} + {0.1056}} \\ {0.6032 - {0.3824}} & {0.1171 - {0.2010}} \\ {0.2610 + {0.1225}} & {{- 0.9157} - {0.2359}} \\ {{- 0.0597} - {0.4649}} & {{- 0.0670} - {0.1213}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},55} \right)} = \begin{matrix} 0.5000 & {0.5295 + {0.2225}} \\ {0.1913 - {0.4619}} & {0.0632 + {0.0938}} \\ {{- 0.3536} - {0.3536}} & {{- 0.3442} + {0.5791}} \\ {{- 0.4619} + {0.1913}} & {0.1494 - {0.4256}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {{- 0.0430} - {0.3791}} & {{- 0.2534} + {0.0743}} \\ {0.5314 - {0.0969}} & {{- 0.0802} + {0.0767}} \\ {0.5001 - {0.1416}} & {{- 0.3092} + {0.6079}} \\ {{- 0.3832} - {0.3817}} & {{- 0.6565} - {0.1479}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},57} \right)} = \begin{matrix} 0.5000 & {{- 0.4933} + {0.2493}} \\ {0.1913 + {0.4619}} & {{- 0.0119} - {0.1919}} \\ {{- 0.3536} + {0.3536}} & {0.2792 + {0.2603}} \\ {{- 0.4619} - {0.1913}} & {{- 0.5616} - {0.4432}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {{- 0.0197} + {0.5406}} & {0.5987 + {0.4276}} \\ {0.1681 + {0.0640}} & {{- 0.2253} + {0.0427}} \\ {0.2143 - {0.3250}} & {{- 0.1122} + {0.5394}} \\ {{- 0.2387} - {0.6830}} & {0.2949 - {0.1252}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},59} \right)} = \begin{matrix} 0.5000 & {{- 0.1299} - {0.1774}} \\ {{- 0.4619} + {0.1913}} & {{- 0.7371} - {0.3712}} \\ {0.3536 - {0.3536}} & {{- 0.1244} - {0.1207}} \\ {{- 0.1913} + {0.4619}} & {0.4090 - {0.2705}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {0.1819 - {0.0589}} & {{- 0.3263} - {0.7161}} \\ {0.3368 - {0.1998}} & {{- 0.0538} + {0.0491}} \\ {0.6565 - {0.0556}} & {0.2749 - {0.2169}} \\ {{- 0.5241} - {0.3183}} & {0.4347 - {0.2526}} \end{matrix}$

${V\; 2\left( {\text{:},\text{:},61} \right)} = \begin{matrix} 0.5000 & {{- 0.3550} + {0.4986}} \\ {{- 0.1913} - {0.4619}} & {{- 0.5979} - {0.4466}} \\ {{- 0.3536} + {0.3536}} & {0.2014 + {0.0225}} \\ {0.4619 + {0.1913}} & {{- 0.1636} - {0.0232}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {0.5307 + {0.2465}} & {0.4315 + {0.2977}} \\ {0.6357 - {0.3878}} & {{- 0.3022} + {0.0611}} \\ {0.1539 - {0.1260}} & {{- 0.4238} - {0.3362}} \\ {{- 0.249} - {0.0328}} & {0.3188 - {0.4857}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},63} \right)} = \begin{matrix} 0.5000 & {0.6135 - {0.5466}} \\ {0.4619 - {0.1913}} & {{- 0.3336} + {0.1681}} \\ {0.3536 - {0.3536}} & {0.0923 + {0.1646}} \\ {0.1913 - {0.4619}} & {0.2389 + {0.3044}} \end{matrix}$ ${V\; 2\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.4886} + {0.2995}} & {{- 0.0643} + {0.2882}} \\ {0.4671 + {0.2039}} & {0.1360 - {0.4277}} \\ {0.5829 + {0.1869}} & {{- 0.4682} + {0.3860}} \\ {{- 0.1465} - {0.1250}} & {{- 0.2986} - {0.5040}} \end{matrix}$

—Final Rank 3 Codebook:

$\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},2} \right)} = \begin{matrix} {{- 0.4516} - {0.0164}} & {{- 0.1786} + {0.5269}} & {{- 0.2985} - {0.2313}} \\ {{- 0.3631} - {0.4030}} & {{- 0.3685} - {0.1544}} & {0.1966 + {0.5090}} \\ {0.4237 + {0.2733}} & {{- 0.5798} - {0.0933}} & {{- 0.1123} - {0.3741}} \\ {0.2522 + {0.4286}} & {{- 0.3685} + {0.2240}} & {0.0615 + {0.6351}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.7071 - {0.3537}} & {{- 0.0971} + {0.0208}} & {{- 0.2518} - {0.2904}} \\ {{- 0.1549} - {0.4387}} & {{- 0.4646} - {0.3323}} & {0.1746 + {0.1101}} \\ {0.3107 - {0.1994}} & {{- 0.1616} + {0.7129}} & {0.4094 + {0.1641}} \\ {{- 0.1065} + {0.1039}} & {{- 0.1624} + {0.3212}} & {{- 0.7503} - {0.2284}} \end{matrix}$ $\mspace{20mu} {{V\; 3\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}}$

${V\; 3\left( {\text{:},\text{:},5} \right)} = \begin{matrix} {0.1911 + {0.1844}} & {{- 0.2742} - {0.7346}} & {0.4333 - {0.3239}} \\ {{- 0.6076} - {0.2220}} & {0.1147 - {0.2232}} & {{- 0.0716} - {0.4327}} \\ {0.0935 - {0.1294}} & {{- 0.2568} - {0.4393}} & {{- 0.3198} + {0.6137}} \\ {0.3520 - {0.6014}} & {0.1308 - {0.2148}} & {{- 0.1887} - {0.0218}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},6} \right)} = \begin{matrix} {{- 0.4390} - {0.0177}} & {0.1890 + {0.1915}} & {0.5399 - {0.1614}} \\ {{- 0.2573} - {0.1623}} & {0.5097 - {0.7079}} & {{- 0.1643} + {0.3299}} \\ {0.4550 + {0.6115}} & {0.1352 - {0.1381}} & {{- 0.0396} - {0.0100}} \\ {0.0477 - {0.3623}} & {{- 0.0601} - {0.3545}} & {0.1275 - {0.7271}} \end{matrix}$ $\mspace{20mu} {{V\; 3\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {{- 0.8432} + {0.1719}} & {0.1018 + {0.2933}} & {0.3119 - {0.2093}} \\ {0.0004 - {0.0643}} & {0.6544 - {0.1493}} & {0.3891 + {0.3966}} \\ {0.0542 + {0.3715}} & {{- 0.5068} + {0.3662}} & {0.2327 + {0.2963}} \\ {{- 0.2612} + {0.2147}} & {0.1621 - {0.1894}} & {{- 0.3150} + {0.5560}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.2307 - {0.3523}} & {{- 0.3946} + {0.2222}} & {0.3873 + {0.4914}} \\ {{- 0.1995} - {0.0992}} & {0.1664 - {0.1501}} & {{- 0.5320} + {0.2829}} \\ {{- 0.6261} + {0.4891}} & {0.1146 + {0.2907}} & {0.0580 + {0.3166}} \\ {{- 0.3749} - {0.0349}} & {{- 0.2411} - {0.7674}} & {0.3196 - {0.1994}} \end{matrix}$ $\mspace{20mu} {{V\; 3\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.3380} - {0.1860}} & {0.2936 - {0.0670}} & {0.1233 + {0.3645}} \\ {{- 0.2536} - {0.2825}} & {{- 0.4027} + {0.1913}} & {{- 0.6695} + {0.1424}} \\ {0.6905 - {0.2237}} & {0.4699 + {0.1884}} & {{- 0.0830} + {0.162}} \\ {{- 0.3771} + {0.1950}} & {0.6721 + {0.0505}} & {{- 0.4095} - {0.4330}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.5870 + {0.5837}} & {0.1694 - {0.3686}} & {{- 0.1020} - {0.0171}} \\ {0.1353 - {0.0302}} & {0.2036 + {0.5782}} & {{- 0.5358} + {0.5295}} \\ {0.1672 - {0.4033}} & {0.4862 - {0.2843}} & {{- 0.3698} - {0.1949}} \\ {0.1619 - {0.2805}} & {0.0806 + {0.3688}} & {0.4237 - {0.2601}} \end{matrix}$

${\mspace{20mu} {{{V\; 3\left( {\text{:},\text{:},13} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}}{{V\; 3\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.2910 + {0.0846}} & {0.0464 + {0.0232}} & {{- 0.8617} + {0.1530}} \\ {{- 0.5838} - {0.6826}} & {0.1164 + {0.0920}} & {{- 0.3062} - {0.1997}} \\ {{- 0.1084} + {0.1034}} & {{- 06815} - {0.4049}} & {0.0000 - {0.01781}} \\ {0.0232 - {0.2799}} & {{- 0.4827} - {0.3376}} & {{- 0.1746} + {0.1956}} \end{matrix}}{{V\; 3 \left( {\text{:}, \text{:}, 15} \right)} =}}\quad}{\quad {{\begin{matrix} {0.4864 + {0.1062}} & {{- 0.5138} + {0.1606}} & {0.6091 + {0.2633}} \\ {{- 0.3579} + {0.0709}} & {0.2824 - {0.4495}} & {0.5834 - {0.0898}} \\ {0.3743 - {0.4217}} & {{- 0.1744} - {0.4396}} & {{- 0.3251} - {0.1626}} \\ {{- 0.2104} + {0.5067}} & {{- 0.1120} + {0.4384}} & {{- 0.2030} - {0.1946}} \end{matrix}\mspace{20mu} V\; 3\left( {\text{:},\text{:},16} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}}}$

${V\; 3\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.3475} + {0.0349}} & {{- 0.5492} + {0.0805}} & {0.4831 - {0.3324}} \\ {{- 0.3183} + {0.2743}} & {0.0020 - {0.0107}} & {0.5026 + {0.1833}} \\ {0.3910 - {0.0682}} & {{- 0.6621} - {0.4101}} & {{- 0.2328} - {0.1188}} \\ {{- 0.5729} + {0.4644}} & {{- 0.2236} - {0.1878}} & {{- 0.5285} + {0.1492}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.3580 + {0.4230}} & {0.2367 + {0.0128}} & {0.0222 - {0.1544}} \\ {{- 0.5016} - {0.3133}} & {0.2038 - {0.2663}} & {{- 0.4319} + {0.4031}} \\ {0.4516 - {0.0120}} & {{- 0.3421} - {0.6404}} & {{- 0.0186} + {0.2825}} \\ {0.2277 - {0.2953}} & {0.5298 + {0.1531}} & {0.5491 + {0.4950}} \end{matrix}$ $\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.500 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}}$

${V\; 3\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {{- 0.0285} - {0.3000}} & {0.2482 - {0.1368}} & {{- 0.3683} + {0.5255}} \\ {{- 0.0146} + {0.4747}} & {0.8444 + {0.1708}} & {0.1466 - {0.0063}} \\ {0.1862 + {0.2754}} & {{- 0.1088} + {0.1084}} & {{- 0.6599} + {0.2244}} \\ {{- 0.4436} - {0.6134}} & {0.3812 - {0.0925}} & {{- 0.1957} - {0.2063}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.4288 + {0.0624}} & {{- 0.4668} + {0.1127}} & {{- 0.4182} + {0.2512}} \\ {{- 0.2155} - {0.5900}} & {0.4068 - {0.1439}} & {{- 0.2850} - {0.2995}} \\ {0.0071 - {0.1770}} & {0.2316 - {0.2993}} & {0.0461 + {0.7582}} \\ {0.1913 + {0.5913}} & {0.6629 + {0.0244}} & {0.0964 + {0.0693}} \end{matrix}$ $\mspace{76mu} {{V\; 3\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {{- 0.0368} - {0.0444}} & {{- 0.1903} + {0.2671}} & {0.4025 - {0.7140}} \\ {0.0931 + {0.1023}} & {0.3098 - {0.6795}} & {{- 0.0892} + {0.0221}} \\ {0.6755 + {0.2582}} & {{- 0.0591} + {0.4709}} & {{- 0.0849} + {0.3313}} \\ {0.1401 + {0.6595}} & {0.3082 - {0.1202}} & {0.4248 - {0.1494}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.5267 + {0.1302}} & {{- 0.3106} - {0.1437}} & {0.7407 - {0.1338}} \\ {{- 0.3082} + {0.1349}} & {{- 0.4197} - {0.5805}} & {{- 0.1306} - {0.1743}} \\ {{- 0.5798} - {0.2378}} & {{- 0.5074} + {0.1972}} & {0.2367 + {0.1351}} \\ {0.4428 + {0.0607}} & {{- 0.2678} + {0.0417}} & {{- 0.5235} + {0.1939}} \end{matrix}$ $\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},25} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.2867} + {0.4568}} & {{- 0.0929} + {0.0656}} & {{- 0.1384} - {0.5281}} \\ {{- 0.3377} + {0.4028}} & {0.6771 - {0.3179}} & {0.2166 + {0.0016}} \\ {{- 0.4367} - {0.4408}} & {{- 0.1123} - {0.3181}} & {{- 0.5156} - {0.3436}} \\ {0.2118 - {0.0548}} & {0.5502 + {0.1048}} & {{- 0.5105} - {0.1021}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.4489 - {0.4294}} & {0.4657 + {0.4116}} & {0.2255 + {0.3897}} \\ {0.0676 - {0.2514}} & {{- 0.5317} - {0.2639}} & {0.1962 + {0.1631}} \\ {{- 0.0562} - {0.3389}} & {{- 0.0047} + {0.3844}} & {{- 0.1563} - {0.6561}} \\ {{- 0.6352} + {0.1576}} & {{- 0.0225} + {0.3362}} & {0.3199 + {0.4182}} \end{matrix}$

$\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},28} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0224 - {0.0857}} & {{- 0.3110} + {0.4440}} & {0.3131 - {0.5095}} \\ {0.4370 + {0.0749}} & {0.5479 + {0.3445}} & {0.3368 + {0.4595}} \\ {0.0927 - {0.6134}} & {{- 0.4211} + {0.0796}} & {{- 0.0820} + {0.4679}} \\ {0.6031 - {0.2167}} & {{- 0.1108} + {0.3020}} & {{- 0.0056} - {0.3033}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {{- 0.2021} + {0.2515}} & {{- 0.1048} + {0.3295}} & {0.4277 + {0.4419}} \\ {{- 0.2703} - {0.8413}} & {0.2200 + {0.2221}} & {{- 0.0633} - {0.0093}} \\ {{- 0.3054} - {0.0715}} & {{- 0.4718} - {0.1404}} & {0.2258 - {0.7007}} \\ {0.1019 + {0.0787}} & {0.3731 - {0.6334}} & {{- 0.2723} - {0.0393}} \end{matrix}$ $\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},31} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 \end{matrix}}$

${V\; 3\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {{- 0.0056} + {0.0131}} & {0.0379 - {0.7861}} & {0.2371 + {0.2967}} \\ {0.0211 + {0.5772}} & {0.4481 - {0.2070}} & {0.0255 - {0.3378}} \\ {0.6155 - {0.2919}} & {0.1617 + {0.2064}} & {{- 0.2637} - {0.0051}} \\ {0.3534 - {0.2780}} & {0.2588 + {0.0366}} & {0.7632 - {0.2983}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.0913 - {0.3481}} & {0.5632 + {0.2802}} & {{- 0.1951} - {0.3100}} \\ {{- 0.5047} + {0.4712}} & {{- 0.0240} - {0.0899}} & {{- 0.2776} - {0.6145}} \\ {0.0947 - {0.5855}} & {{- 0.3176} - {0.3672}} & {{- 0.4474} - {0.0811}} \\ {{- 0.0727} + {0.1915}} & {0.4510 - {0.3955}} & {0.2488 + {0.3775}} \end{matrix}$ $\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},34} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.2602 - {0.3335}} & {0.1575 - {0.6399}} & {0.5258 + {0.2920}} \\ {0.3267 - {0.3041}} & {{- 0.2930} + {0.2159}} & {{- 0.2319} + {0.3158}} \\ {0.1781 + {0.6675}} & {0.1266 - {0.0025}} & {0.4330 - {0.1995}} \\ {{- 0.1799} + {0.3348}} & {{- 0.3623} - {0.5347}} & {{- 0.3289} + {0.3863}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {{- 0.1449} - {0.3470}} & {{- 0.3069} - {0.2742}} & {{- 0.4090} + {0.5345}} \\ {{- 0.0692} + {0.7493}} & {0.0914 + {0.0688}} & {{- 0.3010} - {0.1486}} \\ {{- 0.1504} + {0.4516}} & {0.1108 - {0.0595}} & {{- 0.1821} + {0.5717}} \\ {0.1073 - {0.2330}} & {0.4107 + {0.7956}} & {{- 0.1571} + {0.2230}} \end{matrix}$ $\mspace{85mu} {{V\; 3\left( {\text{:},\text{:},37} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {{- 0.6780} - {0.3635}} & {0.0238 - {0.3004}} & {{- 0.2666} - {0.1906}} \\ {0.1160 + {0.0644}} & {0.7450 + {0.3155}} & {{- 0.4820} + {0.0363}} \\ {{- 0.3301} + {0.2541}} & {0.4517 - {0.0242}} & {0.6746 - {0.3688}} \\ {{- 0.4384} + {0.1574}} & {0.1584 + {0.1580}} & {{- 0.2294} + {0.1232}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {0.2383 + {0.0963}} & {{- 0.0567} - {0.0816}} & {0.8144 + {0.2149}} \\ {0.2810 + {0.0700}} & {{- 0.5121} + {0.5153}} & {{- 0.1356} + {0.3467}} \\ {0.2857 - {0.4528}} & {0.4309 + {0.0872}} & {{- 0.3612} + {0.1144}} \\ {{- 0.4779} + {0.5788}} & {0.3586 + {0.3748}} & {{- 0.0135} - {0.0906}} \end{matrix}$

$\mspace{79mu} {{V\; 3\left( {\text{:},\text{:},40} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000}} \end{matrix}}$ ${V\; 3\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.6361 - {0.0379}} & {0.0303 + {0.1370}} & {{- 0.3832} + {0.4456}} \\ {{- 0.3164} + {0.0577}} & {0.8288 + {0.2462}} & {{- 0.2979} - {0.0876}} \\ {0.0562 + {0.0352}} & {{- 0.1507} - {0.2734}} & {0.0430 - {0.5667}} \\ {0.6782 - {0.1616}} & {0.3675 - {0.0175}} & {0.1704 - {0.4541}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {{- 0.6004} - {0.0993}} & {0.1028 - {0.0939}} & {0.6203 + {0.1250}} \\ {{- 0.4515} - {0.2985}} & {{- 0.0297} + {0.3112}} & {{- 0.5456} - {0.4663}} \\ {{- 0.1174} + {0.2648}} & {{- 0.8756} + {0.2308}} & {{- 0.0070} + {0.2556}} \\ {0.3667 + {0.3441}} & {0.1195 - {0.2205}} & {{- 0.0920} - {0.1031}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},43} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {{- 0.2530} + {0.3209}} & {{- 0.4814} + {0.3797}} & {{- 0.1906} - {0.2722}} \\ {{- 0.2885} - {0.0485}} & {{- 0.4053} - {0.3563}} & {0.1239 + {0.0673}} \\ {{- 0.0415} - {0.7379}} & {{- 0.0707} + {0.5502}} & {0.3560 - {0.0416}} \\ {0.1649 - {0.4171}} & {{- 0.1172} - {0.1069}} & {{- 0.7392} - {0.4413}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.2821 - {0.1279}} & {{- 0.5649} + {0.3170}} & {0.3752 + {0.3388}} \\ {{- 0.0004} - {0.0348}} & {0.5468 - {0.3850}} & {0.6392 + {0.3007}} \\ {{- 0.2923} + {0.0831}} & {{- 0.2695} + {0.2250}} & {0.4521 - {0.0693}} \\ {{- 0.0749} - {0.8971}} & {{- 0.0004} - {0.0994}} & {{- 0.1722} - {0.0816}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},46} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {{- 0.4748} + {0.1205}} & {{- 0.1361} + {0.1678}} & {0.6410 + {0.2967}} \\ {0.2258 + {0.2637}} & {{- 0.0187} - {0.5564}} & {0.4803 + {0.2985}} \\ {{- 0.265} + {0.3289}} & {{- 0.3000} + {0.5816}} & {{- 0.0870} + {0.0961}} \\ {{- 0.2798} + {0.6188}} & {{- 0.1970} - {0.4199}} & {{- 0.3841} - {0.1305}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.0180} - {0.6214}} & {{- 0.4511} + {0.0275}} & {0.2056 + {0.1427}} \\ {{- 0.0262} + {0.4269}} & {0.2038 - {0.1591}} & {{- 0.3709} - {0.0966}} \\ {0.1087 + {0.3811}} & {{- 0.7745} + {0.3487}} & {{- 0.2829} - {0.1551}} \\ {{- 0.5069} + {0.1291}} & {{- 0.0765} - {0.0408}} & {{- 0.2182} + {0.7993}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.0906 - {0.2217}} & {0.1089 - {0.3913}} & {0.6994 + {0.1962}} \\ {{- 0.3471} + {0.3239}} & {{- 0.5370} - {0.3087}} & {{- 0.3246} + {0.1888}} \\ {{- 0.2568} + {0.4202}} & {0.3284 + {0.4249}} & {0.2167 - {0.4149}} \\ {0.6329 - {0.2724}} & {{- 0.3826} + {0.1289}} & {{- 0.0824} - {0.3247}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {{- 0.2542} - {0.0969}} & {0.6056 - {0.2040}} & {{- 0.2829} - {0.4240}} \\ {0.1965 + {0.0675}} & {{- 0.4536} - {0.0227}} & {0.2347 - {0.7163}} \\ {0.8534 + {0.0091}} & {0.3944 + {0.0214}} & {0.0241 + {0.1682}} \\ {0.3805 + {0.0988}} & {{- 0.1788} - {0.4443}} & {{- 0.2841} - {0.2500}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {{- 0.0310} + {0.3127}} & {{- 0.4192} + {0.2451}} & {{- 0.6324} + {0.1246}} \\ {{- 0.4944} - {0.1906}} & {0.1428 + {0.6550}} & {{- 0.0352} + {0.1364}} \\ {0.0394 + {0.1700}} & {0.4123 + {0.2162}} & {{- 0.4882} - {0.5143}} \\ {{- 0.1337} - {0.7565}} & {0.2889 - {0.1209}} & {{- 0.2467} - {0.0313}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.0769 + {0.4899}} & {0.3574 - {0.2210}} & {{- 0.2987} - {0.2809}} \\ {0.0623 - {0.0897}} & {0.7253 - {0.1454}} & {0.4881 + {0.4323}} \\ {{- 0.4402} + {0.1739}} & {{- 0.0113} + {0.3414}} & {0.2179 - {0.2281}} \\ {{- 0.7025} - {0.1568}} & {0.1428 - {0.3730}} & {{- 0.4909} + {0.2576}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {{- 0.1844} + {0.1638}} & {{- 0.6587} + {0.3087}} & {{- 0.1539} - {0.3692}} \\ {0.3389 - {0.3843}} & {{- 0.2949} + {0.3309}} & {0.4206 - {0.3378}} \\ {0.4409 - {0.5467}} & {{- 0.1139} + {0.1683}} & {{- 0.1714} + {0.4313}} \\ {{- 0.1168} - {0.4120}} & {{- 0.2644} - {0.4039}} & {{- 0.5775} - {0.0090}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {{- 0.1461} + {0.1056}} & {0.2135 + {0.2312}} & {{- 0.6745} + {0.4757}} \\ {0.1171 - {0.2010}} & {0.0162 + {0.6417}} & {{- 0.1534} + {0.0178}} \\ {{- 0.9157} - {0.2359}} & {{- 0.0474} - {0.0483}} & {0.1159 - {0.0681}} \\ {{- 0.0670} - {0.1213}} & {{- 0.6245} - {0.3072}} & {{- 0.2668} + {0.4534}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {0.5295 + {0.2225}} & {{- 0.1373} - {0.3127}} & {{- 0.4137} + {0.3638}} \\ {0.0632 + {0.0938}} & {0.1078 + {0.6964}} & {{- 0.4209} - {0.2518{`}}} \\ {{- 0.3442} + {0.5791}} & {{- 0.4500} - {0.0778}} & {{- 0.1048} + {0.2769}} \\ {0.1494 - {0.4256}} & {{- 0.4187} - {0.0536}} & {{- 0.5783} - {0.1840}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {{- 0.2534} + {0.0743}} & {{- 0.3840} + {0.2583}} & {{- 0.2059} - {0.7267}} \\ {{- 0.0802} + {0.0767}} & {0.0963 + {0.7666}} & {0.2344 + {0.2101}} \\ {{- 0.3092} + {0.6079}} & {{- 0.0022} - {0.4149}} & {{- 0.2681} + {0.1439}} \\ {{- 0.6565} - {0.1479}} & {{- 0.1295} + {0.0072}} & {0.1480 + {0.4646}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {{- 0.4933} + {0.2493}} & {{- 0.1093} - {0.6284}} & {{- 0.1925} - {0.0244}} \\ {{- 0.0119} - {0.1919}} & {{- 0.4573} - {0.0414}} & {0.6333 + {0.3179}} \\ {0.2792 + {0.2603}} & {{- 0.2294} - {0.3536}} & {0.0703 - {0.6494}} \\ {{- 0.5616} - {0.4432}} & {{- 0.4520} + {0.0189}} & {{- 0.1228} - {0.1357}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {0.5987 + {0.4276}} & {0.1110 + {0.0636`}} & {0.3849 + {0.0409}} \\ {{- 0.2253} + {0.0427}} & {0.9463 - {0.0868}} & {{- 0.0474} + {0.0990}} \\ {{- 0.1122} + {0.5394}} & {0.0270 + {0.1660}} & {0.0832 - {0.7139}} \\ {0.2949 - {0.1252}} & {0.1828 + {0.1374}} & {0.5310 + {0.1992}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},59} \right)} = \begin{matrix} {{- 0.1299} - {0.1774}} & {0.0635 + {0.6372}} & {{- 0.1454} - {0.5201}} \\ {{- 0.7371} - {0.3712}} & {0.1343 - {0.0486}} & {0.0889 - {0.2013}} \\ {{- 0.1244} - {0.1207}} & {{- 0.2344} - {0.6779}} & {0.0211 - {0.4528}} \\ {0.4090 - {0.2705}} & {0.0763 - {0.2220}} & {{- 0.6105} - {0.2859}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {{- 0.3263} - {0.7161}} & {{- 0.0074} - {0.1928}} & {{- 0.3735} + {0.4091}} \\ {{- 0.0538} + {0.0491}} & {{- 0.7846} + {0.2480}} & {{- 0.2825} - {0.2907}} \\ {0.2749 - {0.2169}} & {0.0684 + {0.2639}} & {0.5764 + {0.1916}} \\ {0.4347 - {0.2526}} & {{- 0.4391} - {0.1363}} & {0.2853 + {0.2800}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {{- 0.3550} + {0.4986}} & {0.2360 - {0.2295}} & {0.3696 + {0.3610}} \\ {{- 0.5979} - {0.4466}} & {{- 0.1015} - {0.0709}} & {0.4087 - {0.1032}} \\ {0.2014 + {0.0225}} & {{- 0.1794} + {0.3659}} & {0.6567 + {0.3341}} \\ {{- 0.1636} - {0.0232}} & {{- 0.8345} + {0.1176}} & {{- 0.0448} - {0.1024}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {0.4315 + {0.2977}} & {0.5279 - {0.2633}} & {{- 0.0096} - {0.1863}} \\ {{- 0.3022} + {0.611}} & {0.0004 + {0.3129}} & {0.4461 + {0.2312}} \\ {{- 0.4238} - {0.3362}} & {0.4544 + {0.2155}} & {{- 0.6006} - {0.2328}} \\ {0.3188 - {0.4857}} & {0.5285 + {0.1480}} & {0.2559 + {0.4820}} \end{matrix}$ ${V\; 3\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {0.6135 - {0.5466}} & {0.1363 + {0.0107}} & {{- 0.0265} - {0.2357}} \\ {{- 0.3336} + {0.1681}} & {0.3619 + {0.5204}} & {0.4467 - {0.0956}} \\ {0.0923 + {0.1646}} & {{- 0.3085} - {0.5415}} & {0.4468 + {0.3555}} \\ {0.2389 + {0.3044}} & {{- 0.1937} + {0.3920}} & {{- 0.5692} + {0.2917}} \end{matrix}$

${V\; 3\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.0643} + {0.2882}} & {0.7408 - {0.0532}} & {{- 0.0130} - {0.1806}} \\ {0.1360 - {0.4277}} & {0.4029 + {0.4160}} & {{- 0.4410} - {0.0947}} \\ {{- 0.4682} + {0.3860}} & {0.0628 + {0.1030}} & {0.4010 - {0.2857}} \\ {{- 0.2986} - {0.5040}} & {{- 0.0741} - {0.3049}} & {{- 0.0166} - {0.7218}} \end{matrix}$

Final Rank 4 Codebook:

${V\; 4\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}$

$\mspace{79mu} {{V\; 4\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 & {0 - {0.5000}} \end{matrix}}$ ${V\; 4\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {{- 0.2573} + {0.5267}} & {{- 0.4516} - {0.0164}} & {{- 0.1786} + {0.5269}} & {{- 0.2985} - {0.2313}} \\ {0.4306 + {0.2510}} & {{- 0.3631} - {0.4030}} & {{- 0.3685} - {0.1544}} & {0.1966 + {0.5090}} \\ {0.4306 + {0.2510}} & {0.4237 + {0.2733}} & {{- 0.5798} - {0.0933}} & {{- 0.1123} - {0.3741}} \\ {{- 0.3995} - {0.0009}} & {0.2522 + {0.4286}} & {{- 0.3685} + {0.2240}} & {0.0615 + {0.6351}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.4182 - {0.2060}} & {0.7071 - {0.3537}} & {{- 0.0971} + {0.0208}} & {{- 0.2518} - {0.2904}} \\ {0.2693 + {0.5849}} & {{- 0.1549} - {0.4387}} & {{- 0.4646} - {0.3323}} & {0.1746 + {0.1101}} \\ {{- 0.3088} + {0.1985}} & {0.3107 - {0.1994}} & {{- 0.1616} + {0.\; 7129}} & {0.4094 + {0.1641}} \\ {{- 0.1743} + {0.4504}} & {{- 0.1065} + {0.1039}} & {{- 0.1624} + {0.3212}} & {{- 0.7503} - {0.2284}} \end{matrix}$

${V\; 4\left( {:{,{:{,9}}}} \right)} = \begin{matrix} {{- 0.1228} + {0.0831}} & {0.1911 + {0.1844}} & {{- 0.2742} - {0.7346}} & {0.4333 - {0.3239}} \\ {0.4892 - {0.2949}} & {{- 0.6076} - {0.2220}} & {0.1147 - {0.2232}} & {{- 0.0716} - {0.4327}} \\ {0.4754 + {0.1031}} & {0.0935 - {0.1294}} & {{- 0.2568} - {0.4393}} & {{- 0.3198} + {0.6137}} \\ {{- 0.3409} - {0.5467}} & {0.3520 - {0.6014}} & {0.1308 - {0.2148}} & {{- 0.1887} - {0.0218}} \end{matrix}$ ${V\; 4\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {0.1663 + {0.6240}} & {{- 0.4390} - {0.0177}} & {0.1890 + {0.1915}} & {0.5399 - {0.1614}} \\ {0.0064 + {0.1030}} & {{- 0.2573} - {0.1623}} & {0.5097 - {0.7079}} & {{- 0.1643} + {0.3299}} \\ {{- 0.3928} + {0.4752}} & {0.4550 + {0.6115}} & {0.1352 - {0.1381}} & {{- 0.0396} - {0.0100}} \\ {{- 0.4732} - {0.0316}} & {0.0477 - {0.3623}} & {{- 0.0601} - {03545}} & {0.1275 - {0.7271}} \end{matrix}$ ${V\; 4\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {0.1228 - {0.0831}} & {{{- 0.0832}} + {0.1719}} & {0.1018 + {0.2933}} & {0.3119 - {0.2093}} \\ {{- 0.4754} - {0.1031}} & {0.0004 - {0.0643}} & {0.6544 - {0.1493}} & {0.3891 + {0.3966}} \\ {{- 0.4892} + {0.2949}} & {0.0542 + {0.3715}} & {{- 0.5068} + {0.3662}} & {0.2327 + {0.2963}} \\ {0.3409 + {0.5467}} & {{- 0.2612} + {0.2147}} & {0.1621 - {0.1894}} & {{- 0.3150} + {0.5560}} \end{matrix}$ ${V\; 4\left( {:{,{:12}}} \right)} = \begin{matrix} {0.4119 - {0.2376}} & {0.2307 - {0.3523}} & {{- 0.3946} + {0.2222}} & {0.3873 + {0.4914}} \\ {0.0073 - {0.7328}} & {{- 0.1995} - {0.0992}} & {0.1664 - {0.1501}} & {{- 0.5320} + {0.2829}} \\ {0.3791 + {0.1546}} & {{- 0.6261} + {0.4891}} & {0.1146 + {0.2907}} & {0.0580 + {0.3166}} \\ {0.2445 - {0.0972}} & {{- 0.3749} - {0.0349}} & {{- 0.2411} - {0.7674}} & {0.3196 - {0.1994}} \end{matrix}$

${V\; 4\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {{- 0.7073} - {0.3349}} & {{- 0.3380} - {0.1860}} & {0.2936 - {0.0670}} & {0.1233 + {0.3645}} \\ {{- 0.2125} + {0.3788}} & {{- 0.2536} - {0.2825}} & {{- 0.4027} + {0.1913}} & {{- 0.6695} + {0.1424}} \\ {{- 0.2125} + {0.3788}} & {0.6905 - {0.2237}} & {0.4699 + {0.1884}} & {{- 0.0830} + {0.1462}} \\ {0.0779 + {0.0648}} & {{- 0.3771} + {0.1950}} & {0.6721 + {0.0505}} & {{- 0.4095} - {0.4330}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.0318} - {0.3722}} & {0.5870 + {0.5837}} & {0.1694 - {03686}} & {{- 0.1020} - {0.0171}} \\ {{- 0.0512} - {0.1869}} & {0.1353 - {0.0302}} & {0.2036 + {0.5782}} & {{- 0.5358} + {0.5295}} \\ {0.5269 + {0.1995}} & {0.1672 - {0.4033}} & {0.4862 - {0.2843}} & {{- 0.3698} - {0.1949}} \\ {0.3031 - {0.6431}} & {0.1619 - {0.2805}} & {0.0806 + {0.3688}} & {0.4237 - {0.2601}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {0.0318 + {0.3722}} & {0.2910 + {0.0846}} & {0.0464 + {0.0232}} & {{- 0.8617} + {0.1530}} \\ {0.1869 - {0.0512}} & {{- 0.5838} - {0.6826}} & {0.1164 + {0.0920}} & {{- 0.3062} - {0.1997}} \\ {0.5269 + {0.1995}} & {{- 0.1084} + {0.1034}} & {{- 0.6815} - {0.4049}} & {{- 0.0000} - {0.1781}} \\ {{- 0.6431} - {0.3031}} & {0.0232 - {0.2799}} & {{- 0.4827} - {0.3376}} & {{- 0.1746} + {0.1956}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {{- 0.1228} + {0.0831}} & {0.4864 + {0.1062}} & {{- 0.5138} + {0.1606}} & {0.6091 + {0.2633}} \\ {{- 0.1031} + {0.4754}} & {{- 0.3579} + {0.0709}} & {0.2824 - {0.4495}} & {0.5834 - {0.0898}} \\ {{- 0.4892} + {0.2949}} & {0.3743 - {0.4217}} & {{- 0.1744} - {0.4396}} & {{- 0.3251} - {0.1626}} \\ {{- 0.5467} + {0.3409}} & {{- 0.2104} + {0.5067}} & {{- 0.1120} + {0.4384}} & {{- 0.2030} - {0.1946}} \end{matrix}$

${V\; 4\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.4119} + {0.2376}} & {{- 0.3475} + {0.0349}} & {{- 0.5492} + {0.0805}} & {0.4831 - {0.3324}} \\ {0.7328 + {0.0073}} & {{- 0.3183} + {0.2743}} & {0.0020 - {0.0107}} & {0.5026 + {0.1833}} \\ {0.3791 + {0.1546}} & {0.3910 - {0.0682}} & {{- 0.6621} - {0.4101}} & {{- 0.2328} - {0.1188}} \\ {{- 0.0972} - {0.2445}} & {{- 0.5729} + {0.4644}} & {{- 0.2236} - {0.1878}} & {{- 0.5285} + {0.1492}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {0.7073 + {0.3349}} & {0.3580 + {0.4230}} & {0.2367 + {0.0128}} & {0.0222 - {0.1544}} \\ {0.3788 + {0.2125}} & {{- 0.5016} - {0.3133}} & {0.2038 - {0.2663}} & {{- 0.4319} + {0.4031}} \\ {{- 0.2125} + {0.3788}} & {0.4516 - {0.0120}} & {{- 0.3421} - {0.6404}} & {{- 0.0186} + {0.2825}} \\ {{- 0.0648} + {0.0779}} & {0.2277 - {0.2953}} & {0.5298 + {0.1531}} & {0.5491 + {0.4950}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},19} \right)} = \begin{matrix} {{- 0.1663} - {0.6240}} & {{- 0.0285} - {0.3000}} & {0.2482 - {0.1368}} & {{- 0.3683} + {0.5255}} \\ {{- 0.1030} + {0.0064}} & {{- 0.0146} + {0.4747}} & {0.8444 + {0.1708}} & {0.1466 - {0.0063}} \\ {{- 0.3928} + {0.4752}} & {0.1862 + {0.2754}} & {{- 0.1088} + {0.1084}} & {{- 0.6599} + {0.2244}} \\ {{- 0.0316} + {0.4372}} & {{- 0.4436} - {0.6134}} & {0.3812 - {0.0925}} & {{- 0.1957} - {0.2063}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {0.2573 - {0.5267}} & {0.4288 + {0.0624}} & {{- 0.4668} + {0.1127}} & {{- 0.4182} + {0.2512}} \\ {0.2510 - {0.4306}} & {{- 0.2155} - {0.5900}} & {0.4068 - {0.1439}} & {{- 0.2850} - {0.29955}} \\ {0.4306 + {0.2510}} & {0.0071 - {0.1770}} & {0.2316 - {0.2993}} & {0.0461 + {0.7582}} \\ {0.0009 - {0.3995}} & {0.1913 + {0.5913}} & {0.6629 + {0.0244}} & {0.0964 + {0.0693}} \end{matrix}$

${V\; 4\left( {:{,{:21}}} \right)} = \begin{matrix} {{- 0.4182} + {0.2060}} & {{- 0.0368} - {0.0444}} & {{- 0.1903} + {0.2671}} & {0.4025 - {0.7140}} \\ {{- 0.5849} + {0.2693}} & {0.0931 + {0.1023}} & {0.3098 - {0.6795}} & {{- 0.0892} + {0.0221}} \\ {{- 0.3088} + {0.1985}} & {0.6755 + {0.2582}} & {{- 0.0591} + {04709}} & {{- 0.0849} + {0.3313}} \\ {0.4504 + {0.1743}} & {01401 + {0.6595}} & {03082 - {0.1202}} & {0.4248 - {0.1494}} \end{matrix}$ ${V\; 4\left( {:{,{:{,22}}}} \right)} = \begin{matrix} {0.1228 - {0.0831}} & {0.5267 + {0.1302}} & {{- 0.3106} - {0.1437}} & {0.7407 - {0.1338}} \\ {{- 0.2949} - {0.4892}} & {{- 0.3082} + {0.1349}} & {{- 0.4197} - {0.5805}} & {{- 0.1306} - {0.1743}} \\ {0.4754 + {0.1031}} & {{- 0.5798} - {0.2378}} & {{- 0.5074} + {0.1972}} & {0.2367 + {0.1351}} \\ {0.5467 - {0.3409}} & {0.4428 + {0.0{.607}}} & {{- 0.2678} + {0.0417}} & {{- 0.5235} + {0.1939}} \end{matrix}$ ${V\; 4\left( {:{,{:{,23}}}} \right)} = \begin{matrix} {{- 0.6051} + {0.1790}} & {{- 0.2867} + {0.4568}} & {{- 0.0929} + {0.0656}} & {{- 0.1384} - {0.5281}} \\ {0.3147 + {0.1351}} & {{- 0.3377} + {0.4028}} & {0.6771 - {0.3179}} & {0.2166 + {0.0016}} \\ {0.3147 + {0.1351}} & {{- 0.4367} - {0.4408}} & {{- 0.1123} - {0.3181}} & {{- 0.5156} - {0.3436}} \\ {{- 0.1801} - {0.5787}} & {0.2118 - {0.0548}} & {0.5502 + {0.1048}} & {{- 0.5105} - {0.1021}} \end{matrix}$ ${V\; 4\left( {:{,{:{,24}}}} \right)} = \begin{matrix} {0.0704 + {0.1417}} & {0.4489 - {0.4294}} & {0.4657 + {0.4116}} & {0.2255 + {0.3897}} \\ {0.1534 + {0.7008}} & {0.0676 - {0.2514}} & {{- 0.5317} - {0.2639}} & {0.1962 + {0.1631}} \\ {{- 0.4248} + {0.3144}} & {{- 0.0562} - {03389}} & {{- 0.0047} + {0.3844}} & {{- 0.1563} - {0.6561}} \\ {{- 0.4053} + {0.1292}} & {{- 0.6352} + {0.1576}} & {{- 0.0225} + {0.3362}} & {0.3199 + {0.4182}} \end{matrix}$

${V\; 4\left( {:{,{:{,25}}}} \right)} = \begin{matrix} {0.5141 + {0.2763}} & {0.0224 - {0.0857}} & {{- 0.3110} + {0.4440}} & {0.3131 - {0.5095}} \\ {0.2189 + {0.1095}} & {0.4370 + {0.0749}} & {0.5479 + {0.3445}} & {0.3368 + {0.4595}} \\ {0.2769 - {0.3593}} & {0.0927 - {0.6134}} & {{- 0.4211} + {0.0796}} & {{- 0.0820} + {0.4679}} \\ {{- 0.6111} - {0.1423}} & {0.6031 - {0.2167}} & {{- 0.1108} + {0.3020}} & {{- 0.0056} - {0.3033}} \end{matrix}$ ${V\; 4\left( {:{,{:{,26}}}} \right)} = \begin{matrix} {{- 0.6051} + {0.1790}} & {{- 0.2021} + {0.2515}} & {{- 0.1048} + {0.3295}} & {0.4277 + {0.4419}} \\ {{- 0.1351} + {0.3147}} & {{- 0.2703} - {0.8413}} & {0.2200 + {0.2221}} & {{- 0.0633} - {0.0093}} \\ {{- 0.3147} - {0.1351}} & {{- 0.3054} - {0.0715}} & {{- 0.4718} - {0.1404}} & {0.2258 - {0.7007}} \\ {{- 0.5787} + {0.1801}} & {0.1019 + {0.0787}} & {0.3731 - {0.6334}} & {{- 0.2723} - {0.0393}} \end{matrix}$ ${V\; 4\left( {:{,{:{,27}}}} \right)} = \begin{matrix} {0.2250 + {0.4308}} & {{- 0.0056} + {0.0131}} & {0.0379 - {0.7861}} & {0.2371 - {0.2967}} \\ {0.3732 - {0.4108}} & {0.0211 + {0.5772}} & {0.4481 - {0.2070}} & {0.0255 - {0.3378}} \\ {0.5913 + {0.2190}} & {0.6155 - {0.2919}} & {0.1617 + {0.2064}} & {{- 0.2637} - {0.0051}} \\ {{- 0.2387} - {0.0328}} & {0.3534 - {0.2780}} & {0.2588 + {0.0366}} & {0.7632 - {0.2983}} \end{matrix}$ ${V\; 4\left( {:{,{:{,28}}}} \right)} = \begin{matrix} {0.5141 + {0.2763}} & {0.0913 - {0.3481}} & {0.5632 + {0.2802}} & {{- 0.1951} - {0.3100}} \\ {{- 0.1095} + {0.2189}} & {{- 0.5047} + {0.4712}} & {{- 0.0240} - {0.0899}} & {{- 0.2776} - {0.6145}} \\ {{- 0.2769} + {0.3593}} & {0.0947 - {0.5855}} & {{- 0.3176} - {0.3672}} & {{- 0.4474} - {0.0811}} \\ {{- 0.1423} + {0.6111}} & {{- 0.0727} + {0.1915}} & {0.4510 - {0.3955}} & {0.2488 + {0.3775}} \end{matrix}$

${V\; 4\left( {:{,{:{,29}}}} \right)} = \begin{matrix} {0.0704 + {0.1417}} & {0.2602 - {0.3335}} & {0.1575 - {0.6399}} & {0.5258 + {0.2920}} \\ {0.7008 - {0.1534}} & {0.3267 - {0.3041}} & {{- 0.2930} + {0.2159}} & {{- 0.2319} + {0.3158}} \\ {0.4248 - {0.3144}} & {0.1781 + {0.6675}} & {0.1266 - {0.0025}} & {0.4330 - {0.1995}} \\ {{- 0.1292} - {0.4053}} & {{- 0.1799} + {0.3348}} & {{- 0.3623} - {0.5347}} & {{- 0.3289} + {0.3863}} \end{matrix}$ ${V\; 4\left( {:{,{:{,30}}}} \right)} = \begin{matrix} {0.2250 + {0.4308}} & {{- 0.1449} - {0.3470}} & {{- 0.3069} - {0.2742}} & {{- 0.4090} + {0.5345}} \\ {0.4108 + {0.3732}} & {{- 0.0692} + {0.7493}} & {0.0914 + {0.0688}} & {{- 0.3010} - {0.1486}} \\ {{- 0.5913} - {0.2190}} & {{- 0.1504} + {0.4516}} & {0.1108 - {0.0595}} & {{- 0.1821} + {0.5717}} \\ {{- 0.0328} + {0.2387}} & {0.1073 - {0.2330}} & {0.4107 + {0.7956}} & {{- 0.1571} + {0.2230}} \end{matrix}$ ${V\; 4\left( {:{,{:{,31}}}} \right)} = \begin{matrix} {{- 0.0376} + {0.4566}} & {{- 0.6780} - {0.3635}} & {0.0238 - {0.3004}} & {{- 0.2666} - {0.1906}} \\ {0.2688 + {0.1481}} & {0.1160 + {0.0644}} & {0.7450 + {0.3155}} & {{- 0.4820} + {0.0363}} \\ {0.1588 - {0.0744}} & {{- 0.3301} + {0.2541}} & {0.4517 - {0.0242}} & {0.6746 - {0.3688}} \\ {{- 0.5917} - {0.5613}} & {{- 0.4384} + {0.1574}} & {0.1584 + {0.1580}} & {{- 0.2294} + {0.1232}} \end{matrix}$ ${V\; 4\left( {:{,{:{,32}}}} \right)} = \begin{matrix} {{- 0.4472} - {0.1208}} & {0.2383 + {0.0963}} & {{- 0.0567} - {0.0816}} & {0.8144 + {0.2149}} \\ {{- 0.0781} + {0.4936}} & {0.2810 + {0.0700}} & {{- 0.5121} + {0.5153}} & {{- 0.1356} + {0.3467}} \\ {{- 0.6133} - {0.0191}} & {0.2857 - {0.4528}} & {0.4309 + {0.0872}} & {{- 0.3612} + {0.1144}} \\ {{- 0.3591} + {0.1738}} & {{- 0.4779} + {0.5788}} & {0.3586 + {0.3748}} & {{- 0.0135} - {0.0906}} \end{matrix}$

${V\; 4\left( {:{,{:{,33}}}} \right)} = \begin{matrix} {0.4535 + {0.1524}} & {0.6361 - {0.0379}} & {0.0303 + {0.1370}} & {{- 0.3832} + {0.4456}} \\ {0.1943 - {0.1220}} & {{- 0.3164} + {0.0577}} & {0.8288 + {0.2462}} & {{- 0.2979} - {0.0876}} \\ {0.7044 - {0.2811}} & {0.0562 + {0.0352}} & {{- 0.1507} - {0.2734}} & {0.0430 - {0.56671}} \\ {{- 0.3738} - {0.0597}} & {0.6782 - {0.1616}} & {0.3675 - {0.0175}} & {0.1704 - {0.4541}} \end{matrix}$ ${V\; 4\left( {:{,{:{,34}}}} \right)} = \begin{matrix} {{- 0.0376} + {0.4566}} & {{- 0.6004} - {0.0993}} & {0.1028 - {0.0939}} & {0.6203 + {0.1250}} \\ {{- 0.1481} + {0.2688}} & {{- 0.4515} - {0.2985}} & {{- 0.0297} + {0.3112}} & {{- 0.5456} - {0.4663}} \\ {{- 0.1588} + {0.0744}} & {{- 0.1174} + {0.2648}} & {{- 0.8756} + {0.2308}} & {{- 0.0070} + {0.2556}} \\ {{- 0.5613} + {0.5917}} & {0.3667 + {0.3441}} & {0.1195 - {0.2205}} & {{- 0.0920} - {0.1031}} \end{matrix}$ ${V\; 4\left( {:{,{:{,35}}}} \right)} = \begin{matrix} {0.2356 + {0.5396}} & {{- 0.2530} + {0.3209}} & {{- 0.4814} + {0.3797}} & {{- 0.1906} - {0.2722}} \\ {0.6510 - {0.4238}} & {{- 0.2885} - {0.0485}} & {{- 0.4053} - {0.3563}} & {0.1239 + {0.0673}} \\ {0.1311 + {0.0167}} & {{- 0.0415} - {0.7379}} & {{- 0.0707} + {0.5502}} & {0.3560 - {0.0416}} \\ {{- 0.0198} - {0.1791}} & {0.1649 - {0.4171}} & {{- 0.1172} - {0.1069}} & {{- 0.7392} - {0.4413}} \end{matrix}$ ${V\; 4\left( {:{,{:{,36}}}} \right)} = \begin{matrix} {0.4535 + {0.1524}} & {0.2821 - {0.1279}} & {{- 0.5649} + {0.3170}} & {0.3753 + {0.3388}} \\ {0.1220 + {0.1943}} & {{- 0.0004} - {0.0348}} & {0.5468 - {0.3850}} & {0.6392 + {0.3007}} \\ {{- 0.7044} + {0.2811}} & {{- 0.2923} + {0.0831}} & {{- 0.2695} + {0.2250}} & {0.4521 - {0.0693}} \\ {{- 0.0597} + {0.3738}} & {{- 0.0749} - {0.8971}} & {{- 0.0004} - {0.0994}} & {{- 0.1722} - {0.0816}} \end{matrix}$

${V\; 4\left( {:{,{:{,37}}}} \right)} = \begin{matrix} {{- 0.4472} - {0.1208}} & {{- 0.4748} + {0.1205}} & {{- 0.1361} + {0.1678}} & {0.6410 + {0.2967}} \\ {0.4936 + {0.0781}} & {0.2258 + {0.2637}} & {{- 0.0187} - {0.5564}} & {0.4803 + {0.2985}} \\ {0.6133 + {0.0191}} & {{- 0.2650} + {0.3289}} & {{- 0.3000} + {0.5816}} & {{- 0.0870} + {0.0961}} \\ {{- 0.1738} - {0.3591}} & {{- 0.2798} + {0.6188}} & {{- 0.1970} - {0.4199}} & {{- 0.3841} - {0.1305}} \end{matrix}$ ${V\; 4\left( {:{,{:{,38}}}} \right)} = \begin{matrix} {0.2356 + {0.5396}} & {{- 0.0180} - {0.6214}} & {{- 0.4511} + {0.0275}} & {0.2056 + {0.1427}} \\ {0.4238 + {0.6510}} & {{- 0.0262} + {0.4269}} & {0.2038 - {0.1591}} & {{- 0.3709} - {0.0966}} \\ {{- 0.1311} - {0.1671}} & {0.1087 + {0.3811}} & {{- 0.7745} + {0.3487}} & {{- 0.2829} - {0.1551}} \\ {{- 0.1791} + {0.0198}} & {{- 0.5069} + {0.1291}} & {{- 0.0765} - {0.0408}} & {{- 0.2182} + {0.7993}} \end{matrix}$ ${V\; 4\left( {:{,{:{,39}}}} \right)} = \begin{matrix} 0.5000 & {0.0906 - {0.2217}} & {0.1089 - {0.3913}} & {0.6994 + {0.1962}} \\ {0.4619 + {0.1913}} & {{- 0.3471} + {0.3239}} & {{- 0.5370} - {0.3087}} & {{- 0.3246} + {0.1888}} \\ {0.3536 + {0.3536}} & {{- 0.2568} + {0.4202}} & {0.3284 + {0.4249}} & {0.2167 - {0.4149}} \\ {0.1913 + {0.4619}} & {0.6329 - {0.2724}} & {{- 0.3826} + {0.1289}} & {{- 0.0824} - {0.3247}} \end{matrix}$ ${V\; 4\left( {:{,{:{,40}}}} \right)} = \begin{matrix} {{- 0.0628} + {0.5038}} & {{- 0.2542} - {0.0969}} & {0.6056 - {0.2040}} & {{- 0.2829} - {0.4240}} \\ {0.3646 + {0.2226}} & {0.1965 + {0.0675}} & {{- 0.4536} - {0.0227}} & {0.2347 - {0.7163}} \\ {0.2517 + {0.1533}} & {0.8534 + {0.0091}} & {0.3944 + {0.0214}} & {0.0241 + {0.1682}} \\ {{- 0.6562} - {0.2058}} & {0.3805 + {0.0988}} & {{- 0.1788} - {0.4443}} & {{- 0.2841} - {0.2500}} \end{matrix}$

${V\; 4\left( {:{,{:{,41}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.0310} + {0.3127}} & {{- 0.4192} + {0.2451}} & {{- 0.6324} + {0.1246}} \\ {{- 0.1913} + {0.4619}} & {{- 0.4944} - {0.1906}} & {0.1428 + {0.6550}} & {{- 0.0352} + {0.1364}} \\ {{- 0.3536} - {0.3536}} & {0.0394 + {0.1700}} & {0.4123 + {0.2162}} & {{- 0.4882} - {0.5143}} \\ {0.4619 - {0.1913}} & {{- 0.1337} - {0.7565}} & {0.2889 - {0.1209}} & {{- 0.2467} - {0.0313}} \end{matrix}$ ${V\; 4\left( {:{,{:{,42}}}} \right)} = \begin{matrix} {0.4184 + {0.4842}} & {0.0769 + {0.4899}} & {0.3574 - {0.2210}} & {{- 0.2987} - {0.2809}} \\ {0.1085 - {0.0629}} & {0.0623 - {0.0897}} & {0.7253 - {0.1454}} & {0.4881 + {0.4323}} \\ {0.5948 - {0.4538}} & {{- 0.4402} + {0.1739}} & {{- 0.0113} + {0.3414}} & {0.2179 - {0.2281}} \\ {{- 0.0601} - {0.1068}} & {{- 0.7025} - {0.1568}} & {0.1428 - {0.3730}} & {{- 0.4909} + {0.2576}} \end{matrix}$ ${V\; 4\left( {:{,{:{,43}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.1844} + {0.1638}} & {{- 0.6587} + {0.3087}} & {{- 0.1539} - {0.3692}} \\ {{- 0.4619} - {0.1913}} & {0.3389 - {0.3843}} & {{- 0.2949} + {0.3309}} & {0.4206 - {0.3378}} \\ {0.3536 + {0.3536}} & {0.4409 - {0.5467}} & {{- 0.1139} + {0.1683}} & {{- 0.1714} + {0.4313}} \\ {{- 0.1913} - {0.4619}} & {{- 0.1168} - {0.4120}} & {{- 0.2644} - {0.4039}} & {{- 0.5775} - {0.0090}} \end{matrix}$ ${V\; 4\left( {:{,{:{,44}}}} \right)} = \begin{matrix} {{- 0.1083} + {0.4189}} & {{- 0.1461} + {0.1056}} & {0.2135 + {0.2312}} & {{- 0.6745} + {0.4757}} \\ {0.6032 - {0.3824}} & {0.1171 - {0.2010}} & {0.0162 + {0.6417}} & {{- 0.1534} + {0.0178}} \\ {0.2610 + {0.1225}} & {{- 0.9157} - {0.2359}} & {{- 0.0474} - {0.0483}} & {0.1159 - {0.0681}} \\ {{- 0.0597} - {0.4649}} & {{- 0.0670} - {0.1213}} & {{- 0.6245} - {0.3072}} & {{- 0.2668} + {0.4534}} \end{matrix}$

${V\; 4\left( {:{,{:{,45}}}} \right)} = \begin{matrix} 0.5000 & {0.5295 + {0.2225}} & {{- 0.1373} - {0.3127}} & {{- 0.4137} + {0.3638}} \\ {0.1913 - {0.4619}} & {0.0632 + {0.0938}} & {0.1078 + {0.6964}} & {{- 0.4209} - {0.2518}} \\ {{- 0.3536} - {0.3536}} & {{- 0.3442} + {0.5791}} & {{- 0.4500} - {0.0778}} & {{- 0.1048} + {0.2769}} \\ {{- 0.4619} + {0.1913}} & {0.1494 - {0.4256}} & {{- 0.4187} - {0.0536}} & {{- 0.5783} - {0.1840}} \end{matrix}$ ${V\; 4\left( {:{,{:{,46}}}} \right)} = \begin{matrix} {{- 0.0430} - {0.3791}} & {{- 0.2534} + {0.0743}} & {{- 0.3840} + {0.2583}} & {{- 0.2059} - {0.7267}} \\ {0.5314 - {0.0969}} & {{- 0.0802} + {0.0767}} & {0.0963 + {0.7666}} & {0.2344 + {0.2101}} \\ {0.5001 - {0.1416}} & {{- 0.3092} + {0.6079}} & {{- 0.0022} - {0.4149}} & {{- 0.2681} + {0.1439}} \\ {{- 0.3832} - {0.3817}} & {{- 0.6565} - {0.1479}} & {{- 0.1295} + {0.0072}} & {0.1480 + {0.4646}} \end{matrix}$ ${V\; 4\left( {:{,{:{,47}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.4933} + {0.2493}} & {{- 0.1093} - {0.6284}} & {{- 0.1925} - {0.0244}} \\ {0.1913 + {0.4619}} & {{- 0.0119} - {0.1919}} & {{- 0.4573} - {0.0414}} & {0.6333 + {0.3179}} \\ {{- 0.3536} + {0.3536}} & {0.2792 + {0.2603}} & {{- 0.2294} - {0.3536}} & {0.0703 - {0.6494}} \\ {{- 0.4619} - {0.1913}} & {{- 0.5616} - {0.4432}} & {{- 0.4520} + {0.0189}} & {{- 0.1228} - {0.1357}} \end{matrix}$ ${V\; 4\left( {:{,{:{,48}}}} \right)} = \begin{matrix} {{- 0.0197} + {0.5406}} & {0.5987 + {0.4276}} & {0.1110 + {0.0636}} & {0.3849 + {0.0409}} \\ {0.1681 + {0.0640}} & {{- 0.2253} + {0.0427}} & {0.9463 - {0.0868}} & {{- 0.0474} + {0.0990}} \\ {0.2143 - {0.3250}} & {{- 0.1122} + {0.5394}} & {0.0270 + {0.1660}} & {0.0832 - {0.7139}} \\ {{- 0.2387} - {0.6830}} & {0.2949 - {0.1252}} & {0.1828 + {0.1374}} & {0.5310 + {0.1992}} \end{matrix}$

${V\; 4\left( {:{,{:{,49}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.1299} - {0.1774}} & {0.0635 + {0.6372}} & {{- 0.1454} - {0.5201}} \\ {{- 0.4619} + {0.1913}} & {{- 0.7371} - {0.3712}} & {0.1343 - {0.0486}} & {0.0889 - {0.2013}} \\ {0.3536 - {0.3536}} & {{- 0.1244} - {0.1207}} & {{- 0.2344} - {0.6779}} & {0.0211 - {0.4528}} \\ {{- 0.1913} + {0.4619}} & {0.4090 - {0.2705}} & {0.0763 - {0.2220}} & {{- 0.6105} - {0.2859}} \end{matrix}$ ${V\; 4\left( {:{,{:{,50}}}} \right)} = \begin{matrix} {0.1819 - {0.0589}} & {{- 0.3263} - {0.7161}} & {{- 0.0074} - {0.1928}} & {{- 0.3735} + {0.4091}} \\ {0.3368 - {0.1998}} & {{- 0.0538} + {0.0491}} & {{- 0.7846} + {0.2480}} & {{- 0.2825} - {0.2907}} \\ {0.6565 - {0.0556}} & {0.2749 - {0.2169}} & {0.0684 + {0.2639}} & {0.5764 + {0.1916}} \\ {{- 0.5241} - {0.3183}} & {0.4347 - {0.2526}} & {{- 0.4391} - {0.1363}} & {0.2853 + {0.2800}} \end{matrix}$ ${V\; 4\left( {:{,{:{,51}}}} \right)} = \begin{matrix} 0.5000 & {{- 0.3550} + {0.4986}} & {0.2360 - {0.2295}} & {0.3696 + {0.3610}} \\ {{- 0.1913} - {0.4619}} & {{- 0.5979} - {0.4466}} & {{- 0.1015} - {0.0709}} & {0.4087 - {0.1032}} \\ {{- 0.3536} + {0.3536}} & {0.2014 + {0.0225}} & {{- 0.1794} + {0.3659}} & {0.6567 + {0.3341}} \\ {0.4619 + {0.1913}} & {{- 0.1636} - {0.0232}} & {{- 0.8345} + {0.1176}} & {{- 0.0448} - {0.1024}} \end{matrix}$ ${V\; 4\left( {:{,{:{,52}}}} \right)} = \begin{matrix} {0.5307 + {0.2465}} & {0.4315 + {0.2977}} & {0.5279 - {0.2633}} & {{- 0.0096} - {0.1863}} \\ {0.6357 - {0.3878}} & {{- 0.3022} + {0.0611}} & {0.0004 + {0.3129}} & {0.4461 + {0.2312}} \\ {0.1539 - {0.1260}} & {{- 0.4238} - {0.3362}} & {0.4544 + {0.2155}} & {{- 0.6006} - {0.2328}} \\ {{- 0.2499} - {0.0328}} & {0.3188 - {0.4857}} & {0.5285 + {0.1480}} & {0.2559 + {0.4820}} \end{matrix}$

${V\; 4\left( {:{,{:{,53}}}} \right)} = \begin{matrix} 0.5000 & {0.6135 - {0.5466}} & {0.1363 + {0.0107}} & {{- 0.0265} - {0.2357}} \\ {0.4619 - {0.1913}} & {{- 0.3336} + {0.1681}} & {0.3619 + {0.5204}} & {0.4467 - {0.0956}} \\ {0.3536 - {0.3536}} & {0.0923 + {0.1646}} & {{- 0.3085} - {0.5415}} & {0.4468 + {0.3555}} \\ {0.1913 - {0.4619}} & {0.2389 + {0.3044}} & {{- 0.1937} + {0.3920}} & {{- 0.5692} + {0.2917}} \end{matrix}$ ${V\; 4\left( {:{,{:{,54}}}} \right)} = \begin{matrix} {{- 0.4886} + {0.2995}} & {{- 0.0643} + {0.2882}} & {0.7408 - {0.0532}} & {{- 0.0130} - {0.1806}} \\ {0.4671 + {0.2039}} & {0.1360 - {0.4277}} & {0.4029 + {0.4160}} & {{- 0.4410} - {0.0947}} \\ {0.5829 + {0.1869}} & {{- 0.4682} + {0.3860}} & {0.0628 + {0.1030}} & {0.4010 - {0.2857}} \\ {{- 0.1465} - {0.1250}} & {{- 0.2986} - {0.5040}} & {{- 0.0741} - {0.3049}} & {{- 0.0166} - {0.7218}} \end{matrix}$

The 1^(st), 4^(th), 7^(th), 10^(th), 13^(th), 16^(th), 19^(th), 22^(nd), 25^(th), 28^(th), 31^(st), 34^(th), 37^(th), 40^(th), 43^(rd), and 46^(th) codewords in each of the 6-bit final rank 1 codebook, the final rank 2 codebook, and the final rank 3 codebook are equal to the 16 codewords included in each of the 4-bit rank codebook, the rank 2 codebook, and the rank 3 codebook, disclosed in the above Table 1.

The 6-bit final rank 1 codebook, the final rank 2 codebook, and the final rank 3 codebook may be obtained using column subsets of the rank 4 codebook. Accordingly, the 6-bit final rank 1 codebook, the final rank 2 codebook, the final rank 3 codebook, and the final rank 4 codebook may be expressed as a function of V4, as given by the following Table 14, and may also be expressed using various types of schemes:

Transmit Trans- Codebook mission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 V4(:,2,1) V4(:,[1,2],1) V4(:,[1,2,3],1) V4(:,:,1) 2 V4(:,1,7) V4(:,[1,2],7) V4(:,[2,3,4],7) V4(:,:,2) 3 V4(:,1,8) V4(:,[1,2],8) V4(:,[2,3,4],8) V4(:,:,3) 4 V4(:,3,1) V4(:,[1,3],1) V4(:,[1,2,4],1) V4(:,:,4) 5 V4(:,1,9) V4(:,[1,2],9) V4(:,[2,3,4],9) V4(:,:,5) 6 V4(:,1,10) V4(:,[1,2],10) V4(:,[2,3,4],10) V4(:,:,6) 7 V4(:,4,1) V4(:,[1,4],1) V4(:,[1,3,4],1) V4(:,:,7) 8 V4(:,1,11) V4(:,[1,2],11) V4(:,[2,3,4],11) V4(:,:,8) 9 V4(:,1,12) V4(:,[1,2],12) V4(:,[2,3,4],12) V4(:,:,9) 10 V4(:,2,2) V4(:,[2,3],1) V4(:,[2,3,4],1) V4(:,:,10) 11 V4(:,1,13) V4(:,[1,2],13) V4(:,[2,3,4],13) V4(:,:,11) 12 V4(:,1,14) V4(:,[1,2],14) V4(:,[2,3,4],14) V4(:,:,12) 13 V4(:,3,2) V4(:,[2,4],1) V4(:,[1,2,3],2) V4(:,:,13) 14 V4(:,1,15) V4(:,[1,2],15) V4(:,[2,3,4],15) V4(:,:,14) 15 V4(:,1,16) V4(:,[1,2],16) V4(:,[2,3,4],16) V4(:,:,15) 16 V4(:,4,2) V4(:,[3,4],1) V4(:,[1,2,4],2) V4(:,:,16) 17 V4(:,1,17) V4(:,[1,2],17) V4(:,[2,3,4],17) V4(:,:,17) 18 V4(:,1,18) V4(:,[1,2],18) V4(:,[2,3,4],18) V4(:,:,18) 19 V4(:,1,3) V4(:,[1,3],2) V4(:,[1,3,4],2) V4(:,:,19) 20 V4(:,1,19) V4(:,[1,2],19) V4(:,[2,3,4],19) V4(:,:,20) 21 V4(:,1,20) V4(:,[1,2],20) V4(:,[2,3,4],20) V4(:,:,21) 22 V4(:,1,4) V4(:,[1,4],2) V4(:,[2,3,4],2) V4(:,:,22) 23 V4(:,1,21) V4(:,[1,2],21) V4(:,[2,3,4],21) V4(:,:,23) 24 V4(:,1,22) V4(:,[1,2],22) V4(:,[2,3,4],22) V4(:,:,24) 25 V4(:,1,5) V4(:,[2,3],2) V4(:,[1,2,3],3) V4(:,:,25) 26 V4(:,1,23) V4(:,[1,2],23) V4(:,[2,3,4],23) V4(:,:,26) 27 V4(:,1,24) V4(:,[1,2],24) V4(:,[2,3,4],24) V4(:,:,27) 28 V4(:,2,5) V4(:,[2,4],2) V4(:,[1,3,4],3) V4(:,:,28) 29 V4(:,1,25) V4(:,[1,2],25) V4(:,[2,3,4],25) V4(:,:,29) 30 V4(:,1,26) V4(:,[1,2],26) V4(:,[2,3,4],26) V4(:,:,30) 31 V4(:,3,5) V4(:,[1,3],3) V4(:,[1,2,3],4) V4(:,:,31) 32 V4(:,1,27) V4(:,[1,2],27) V4(:,[2,3,4],27) V4(:,:,32) 32 V4(:,1,28) V4(:,[1,2],28) V4(:,[2,3,4],28) V4(:,:,33) 34 V4(:,4,5) V4(:,[1,4],3) V4(:,[1,3,4],4) V4(:,:,34) 35 V4(:,1,29) V4(:,[1,2],29) V4(:,[2,3,4],29) V4(:,:,35) 36 V4(:,1,30) V4(:,[1,2],30) V4(:,[2,3,4],30) V4(:,:,36) 37 V4(:,1,6) V4(:,[1,3],4) V4(:,[1,2,3],5) V4(:,:,37) 38 V4(:,1,31) V4(:,[1,2],31) V4(:,[2,3,4],31) V4(:,:,38) 39 V4(:,1,32) V4(:,[1,2],32) V4(:,[2,3,4],32) V4(:,:,39) 40 V4(:,2,6) V4(:,[1,4],4) V4(:,[1,3,4],5) V4(:,:,40) 41 V4(:,1,33) V4(:,[1,2],33) V4(:,[2,3,4],33) V4(:,:,41) 42 V4(:,1,34) V4(:,[1,2],34) V4(:,[2,3,4],34) V4(:,:,42) 43 V4(:,3,6) V4(:,[1,3],5) V4(:,[1,2,4],6) V4(:,:,43) 44 V4(:,1,35) V4(:,[1,2],35) V4(:,[2,3,4],35) V4(:,:,44) 45 V4(:,1,36) V4(:,[1,2],36) V4(:,[2,3,4],36) V4(:,:,45) 46 V4(:,4,6) V4(:,[2,4],6) V4(:,[2,3,4],6) V4(:,:,46) 47 V4(:,1,37) V4(:,[1,2],37) V4(:,[2,3,4],37) V4(:,:,47) 48 V4(:,1,38) V4(:,[1,2],38) V4(:,[2,3,4],38) V4(:,:,48) 49 V4(:,1,39) V4(:,[1,2],39) V4(:,[2,3,4],39) V4(:,:,49) 50 V4(:,1,40) V4(:,[1,2],40) V4(:,[2,3,4],40) V4(:,:,50) 51 V4(:,1,41) V4(:,[1,2],41) V4(:,[2,3,4],41) V4(:,:,51) 52 V4(:,1,42) V4(:,[1,2],42) V4(:,[2,3,4],42) V4(:,:,52) 53 V4(:,1,43) V4(:,[1,2],43) V4(:,[2,3,4],43) V4(:,:,53) 54 V4(:,1,44) V4(:,[1,2],44) V4(:,[2,3,4],44) V4(:,:,54) 55 V4(:,1,45) V4(:,[1,2],45) V4(:,[2,3,4],45) 56 V4(:,1,46) V4(:,[1,2],46) V4(:,[2,3,4],46) 57 V4(:,1,47) V4(:,[1,2],47) V4(:,[2,3,4],47) 58 V4(:,1,48) V4(:,[1,2],48) V4(:,[2,3,4],48) 59 V4(:,1,49) V4(:,[1,2],49) V4(:,[2,3,4],49) 60 V4(:,1,50) V4(:,[1,2],50) V4(:,[2,3,4],50) 61 V4(:,1,51) V4(:,[1,2],51) V4(:,[2,3,4],51) 62 V4(:,1,52) V4(:,[1,2],52) V4(:,[2,3,4],52) 63 V4(:,1,53) V4(:,[1,2],53) V4(:,[2,3,4],53) 64 V4(:,1,54) V4(:,[1,2],54) V4(:,[2,3,4],54)

4. Fourth Scheme to Design a 6-Bit Codebook

(1) Operation 1:

The following 4-bit codebook corresponding to the transmission rank 1 may be obtained from the 4-bit codebook disclosed in the above Table 1. Here, the 4-bit codebook corresponding to the transmission rank 1 disclosed in the above Table 1 is referred to as a base codebook.

base_cbk(:,1:16)=[C _(1,1) . . . C _(16,1)] with C _(i,1) taken from table 1

(2) Operation 2:

A local codebook local_cbk may be defined, for example, as follows:

${{local\_ cbk} = \begin{bmatrix} \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975} \end{matrix} & \begin{matrix} {{- 0.0975} -} \\ {0.4904} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975} \end{matrix} & \begin{matrix} {0.0975 +} \\ {0.4904} \end{matrix} \\ \begin{matrix} {{- 0.4904} +} \\ {0.0975} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975} \end{matrix} & \begin{matrix} {{- 0.4904} +} \\ {0.0975} \end{matrix} & \begin{matrix} {0.4904 -} \\ {0.0975} \end{matrix} \\ \begin{matrix} {0.3536 +} \\ {0.3536} \end{matrix} & \begin{matrix} {0.3536 -} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} -} \\ {0.3536} \end{matrix} & \begin{matrix} {{- 0.3536} +} \\ {0.3536} \end{matrix} \end{bmatrix}};$

(3) Operation 3:

r=abs(local_cbk); phase_local_cbk=local_cbk./r; alpha=0.9835;

A localized codebook localized_cbk may be expressed, for example, as follows:

localized_cbk= [sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,1)){circumflex over ( )}2))*phase_local_cbk(1,1),sqrt(1−alpha{circumflex over ( )}2*(1− (r(1,2)){circumflex over ( )}2))*phase_local_cbk(1,2),sqrt(1−alpha{circumflex over ( )}2*(1−(r(1,3)){circumflex over ( )}2))*phase_local_cbk(1,3),sqrt(1− alpha{circumflex over ( )}2*(1−r(1,4)){circumflex over ( )}2))*phase_local_cbk(1,4);... alpha*r(2,1)*phase_local_cbk(2,1),alpha*r(2,2)*phase_local_cbk(2,2),alpha*r(2,3)*phase_local_cb k(2,3),alpha*r(2,4)*phase_local_cbk(2,4);... alpha*r(3,1)*phase_local_cbk(3,1),alpha*r(3,2)*phase_local_cbk(3,2),alpha*r(3,3)*phase_local_cb k(3,3),alpha*r(3,4)*phase_local_cbk(3,4);... alpha*r(4,1)*phase_local_cbk(4,1),alpha*r(4,2)*phase_local_cbk(4,2),alpha*r(4,3)*phase_local_cb k(4,3),alpha*r(4,4)*phase_local_cbk(4,4)];

Where the localized codebook localized_cbk is calculated through the aforementioned process, vectors of the localized codebook localized_cbk may be normalized, for example, through the following exemplary process:

for k=1:size(localized_cbk,2) localized_cbk(:,k)=localized_cbk(:,k)/norm(localized_cbk(:,k)); end

(4) Operation 4:

A final codebook final_cbk may be obtained by rotating the normalized localized codebook localized_cbk around the base codebook, for example, through the following exemplary process:

for k=1:16 [U,S,V]=svd(base_cbk(:,k)); %, where an SVD is performed with respect to the elements of the base codebook. R=U′*V; % U′*V*base_cbk(:,1,k)=[1;0;0;0]. % , where R denotes a rotation matrix that rotates the normalized localized codebook localized_cbk around the base codebook. rotated_localized=R′*localized_cbk(:,1:3);%, where only first three vectors of localized_cbk are rotated. final_cbk(:,(k−1)*4+1:k*4)=[base_cbk(:,k),rotated_localized];%, where the base codebook is maintained as a centroid and the base codebook is included in a final 6-bit codebook. end;

Six bits of the final rank 1 codebook may be given by final_cbk. A final rank 2 codebook, a final rank 3 codebook, and a final rank 4 codebook may also be obtained based on the final rank 1 codebook. For example, unitary matrices including columns of the final rank 1 codebook, different from the first 16 vectors of the base codebook, may be obtained. A total of 48 unitary matrices may be obtained. A total of 54 matrices may be obtained including W₁ through W₆, and the 48 unitary matrices.

The final rank 2 codebook may be obtained by taking the first two columns from the 48 unitary matrices, and by taking 16 matrices or codewords from the 4-bit rank 2 codebook disclosed in the above Table 1.

The final rank 3 codebook may be obtained by taking the second through the fourth columns from the 48 unitary matrices, and by taking 16 matrices or codewords from the 4-bit rank 3 codebook disclosed in the above Table 1.

Six bits of the final rank 1 codebook, the final rank 2 codebook, the final rank 3 codebook, and the final rank 4 codebook may be expressed as follows:

Final Rank 1 Codebook:

${V\; 1\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {:{,{:{,2}}}} \right)} = \begin{matrix} {0.3260 + {0.6774}} \\ {0.3254 + {0.1709}} \\ {0.3254 + {0.1709}} \\ {{- 0.0250} + {0.4051}} \end{matrix}$ ${V\; 1\left( {:{,{:{,3}}}} \right)} = \begin{matrix} {0.1499 + {0.0{.347}}} \\ {0.5009 + {0.3071}} \\ {0.1505 + {0.5412}} \\ {0.1505 + {0.5412}} \end{matrix}$ ${V\; 1\left( {:{,{:{,4}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} \\ {0.1311 + {0.6387}} \\ {0.3473 + {0.0541}} \\ {0.4815 + {0.4045}} \end{matrix}$

${V\; 1\left( {:{,{:{,5}}}} \right)} = \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {:{,{:{,6}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} \\ {{- 0.1311} - {0.6387}} \\ {0.2473 + {0.0541}} \\ {{- 0.4815} - {0.4045}} \end{matrix}$ ${V\; 1\left( {:{,{:{,7}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} \\ {0.0056 - {0.3076}} \\ {0.2285 + {0.6580}} \\ {{- 0.3448} - {0.0735}} \end{matrix}$ ${V\; 1\left( {:{,{:{,8}}}} \right)} = \begin{matrix} {0.3260 + {0.6774}} \\ {{- 0.3254} - {0.1709}} \\ {0.3254 + {0.1709}} \\ {0.0250 - {0.4051}} \end{matrix}$

${V\; 1\left( {:{,{:{,9}}}} \right)} = \begin{matrix} {- 0.5000} \\ {- 0.5000} \\ 0.5000 \\ 0.5000 \end{matrix}$ ${V\; 1\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} \\ {{- 0.2473} - {0.0541}} \\ {0.1311 + {0.6387}} \\ {0.4815 + {0.4045}} \end{matrix}$ ${V\; 1\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {{- 0.0337} - {0.6193}} \\ {{- 0.4621} - {0.5019}} \\ {0.2280 + {0.1515}} \\ {0.2280 + {0.1515}} \end{matrix}$ ${V\; 1\left( {:{,{:{,12}}}} \right)} = \begin{matrix} {{- 0.4422} - {0.0928}} \\ {{- 0.2479} - {0.5606}} \\ {0.2479 + {0.5606}} \\ {0.0137 + {0.2102}} \end{matrix}$

${V\; 1\left( {:{,{:{,13}}}} \right)} = \begin{matrix} {- 0.5000} \\ 0.5000 \\ 0.5000 \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {:{,{:{,14}}}} \right)} = \begin{matrix} {{- 0.4422} - {0.0928}} \\ {0.2479 + {0.5606}} \\ {0.2479 + {0.5606}} \\ {{- 0.0137} - {0.2102}} \end{matrix}$ ${V\; 1\left( {:{,{:{,15}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} \\ {{- 0.0056} + {0.3076}} \\ {0.3448 + {0.0735}} \\ {{- 0.2285} - {0.6580}} \end{matrix}$ ${V\; 1\left( {:{,{:{,16}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} \\ {0.2473 + {0.0541}} \\ {0.1311 + {0.6387}} \\ {{- 0.4815} - {0.4045}} \end{matrix}$

${V\; 1\left( {:{,{:{,17}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ 0.5000 \end{matrix} \\ {0 + {0.5000}} \end{matrix}$ ${V\; 1\left( {:{,{:{,18}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {{- 0.3076} - {0.0056}} \\ {0.3448 + {0.0735}} \end{matrix} \\ {{- 0.6580} + {0.2285}} \end{matrix}$ ${V\; 1\left( {:{,{:{,19}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270}} \\ {{- 0.0541} + {0.2473}} \\ {0.1311 + {0.6387}} \end{matrix} \\ {{- 0.4045} + {0.4815}} \end{matrix}$ ${V\; 1\left( {:{,{:{,20}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193}} \\ {{- 0.5019} + {0.4621}} \\ {0.2280 + {0.1515}} \end{matrix} \\ {{- 0.1515} + {0.2280}} \end{matrix}$

${V\; 1\left( {:{,{:{,21}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ 0.5000 \end{matrix} \\ {0 - {0.5000}} \end{matrix}$ ${V\; 1\left( {:{,{:{,22}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193}} \\ {0.5019 - {0.4621}} \\ {0.2280 + {0.1515}} \end{matrix} \\ {0.1515 - {0.2280}} \end{matrix}$ ${V\; 1\left( {:{,{:{,23}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928}} \\ {0.5606 - {0.2479}} \\ {0.2479 + {0.5606}} \end{matrix} \\ {0.2102 - {0.0137}} \end{matrix}$ ${V\; 1\left( {:{,{:{,24}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {0.3076 + {0.0056}} \\ {0.3448 + {0.0735}} \end{matrix} \\ {0.6580 - {0.2285}} \end{matrix}$

${V\; 1\left( {:{,{:{,25}}}} \right)} = \begin{matrix} \begin{matrix} {- 0.5000} \\ {0 - {0.5000}} \\ 0.5000 \end{matrix} \\ {0 + {0.5000}} \end{matrix}$ ${V\; 1\left( {:{,{:{,26}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.3841} - {0.3851}} \\ {0.3076 + {0.0056}} \\ {0.2285 + {0.6580}} \end{matrix} \\ {{- 0.0735} + {0.3448}} \end{matrix}$ ${V\; 1\left( {:{,{:{,27}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.3260} - {0.6774}} \\ {0.1709 - {0.3254}} \\ {0.3254 + {0.1709}} \end{matrix} \\ {{- 0.4051} - {0.0250}} \end{matrix}$ ${V\; 1\left( {:{,{:{,28}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1499} - {0.0347}} \\ {0.3017 - {0.5009}} \\ {0.1505 + {0.5412}} \end{matrix} \\ {{- 0.5412} + {0.1505}} \end{matrix}$

${V\; 1\left( {:{,{:{,29}}}} \right)} = \begin{matrix} \begin{matrix} {- 0.5000} \\ {0 + {0.5000}} \\ 0.5000 \end{matrix} \\ {0 - {0.5000}} \end{matrix}$ ${V\; 1\left( {:{,{:{,30}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1499} - {0.0347}} \\ {{- 0.3071} + {0.5009}} \\ {0.1505 + {0.5412}} \end{matrix} \\ {0.5412 - {0.1505}} \end{matrix}$ ${V\; 1\left( {:{,{:{,31}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.0918} - {0.3270}} \\ {{- 0.6387} + {0.1311}} \\ {0.2473 + {0.0541}} \end{matrix} \\ {0.4045 - {0.4815}} \end{matrix}$ ${V\; 1\left( {:{,{:{,32}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.3841} - {0.3851}} \\ {{- 0.3076} - {0.0056}} \\ {0.2285 + {0.6580}} \end{matrix} \\ {0.0735 - {0.3448}} \end{matrix}$

${V\; 1\left( {:{,{:{,33}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix} \\ {- 0.5000} \end{matrix}$ ${V\; 1\left( {:{,{:{,34}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193}} \\ {0.2280 + {0.1515}} \\ {0.2280 + {0.1515}} \end{matrix} \\ {{- 0.4621} - {0.5019}} \end{matrix}$ ${V\; 1\left( {:{,{:{,35}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270}} \\ {0.4815 + {0.4045}} \\ {0.1311 + {0.6387}} \end{matrix} \\ {{- 0.2473} + {0.0541}} \end{matrix}$ ${V\; 1\left( {:{,{:{,36}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {0.2285 + {0.6580}} \\ {0.3448 + {0.0735}} \end{matrix} \\ {0.0056 - {0.3076}} \end{matrix}$

${V\; 1\left( {:{,{:{,37}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 + {0.5000}} \\ {- 0.5000} \end{matrix} \\ {0 + {0.5000}} \end{matrix}$ ${V\; 1\left( {:{,{:{,38}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928}} \\ {{- 0.2102} + {0.0137}} \\ {{- 0.2479} - {0.5606}} \end{matrix} \\ {{- 0.5606} + {0.2479}} \end{matrix}$ ${V\; 1\left( {:{,{:{,39}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193}} \\ {{- 0.1515} + {0.2280}} \\ {{- 0.2280} - {0.1515}} \end{matrix} \\ {{- 0.5019} + {0.4621}} \end{matrix}$ ${V\; 1\left( {:{,{:{,40}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270}} \\ {{- 0.4045} + {0.4815}} \\ {{- 0.1311} - {0.6387}} \end{matrix} \\ {{- 0.0541} + {0.2473}} \end{matrix}$

${V\; 1\left( {:{,{:{,41}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {- 0.5000} \\ 0.5000 \end{matrix} \\ 0.5000 \end{matrix}$ ${V\; 1\left( {:{,{:{,42}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {{- 0.2285} - {0.6580}} \\ {0.3448 + {0.0735}} \end{matrix} \\ {{- 0.0056} + {0.3076}} \end{matrix}$ ${V\; 1\left( {:{,{:{,43}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928}} \\ {{- 0.0137} - {0.2102}} \\ {0.2479 + {0.5606}} \end{matrix} \\ {0.2470 + {0.5606}} \end{matrix}$ ${V\; 1\left( {:{,{:{,44}}}} \right)} = \begin{matrix} \begin{matrix} {0.0337 + {0.6193}} \\ {{- 0.2280} - {0.1515}} \\ {0.2280 + {0.1515}} \end{matrix} \\ {0.4621 + {0.5019}} \end{matrix}$

${V\; 1\left( {:{,{:{,45}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0 - {0.5000}} \\ {- 0.5000} \end{matrix} \\ {0 - {0.5000}} \end{matrix}$ ${V\; 1\left( {:{,{:{,46}}}} \right)} = \begin{matrix} \begin{matrix} {0.0918 + {0.3270}} \\ {0.4045 - {0.4815}} \\ {{- 0.1311} - {0.6387}} \end{matrix} \\ {0.0541 - {0.2473}} \end{matrix}$ ${V\; 1\left( {:{,{:{,47}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {0.6580 - {0.2285}} \\ {{- 0.3448} - {0.0735}} \end{matrix} \\ {0.3076 + {0.0056}} \end{matrix}$ ${V\; 1\left( {:{,{:{,48}}}} \right)} = \begin{matrix} \begin{matrix} {0.4422 + {0.0928}} \\ {0.2102 - {0.0137}} \\ {{- 0.2479} - {0.5606}} \end{matrix} \\ {0.5606 - {0.2479}} \end{matrix}$

${V\; 1\left( {:{,{:{,49}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0 + {0.5000}} \end{matrix} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ ${V\; 1\left( {:{,{:{,50}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {0.0022 + {0.1690}} \\ {{- 0.3076} - {0.0056}} \end{matrix} \\ {{- 0.7536} - {0.1140}} \end{matrix}$ ${V\; 1\left( {:{,{:{,51}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1560} + {0.4926}} \\ {0.0837 + {0.4175}} \\ {{- 0.4597} + {0.3989}} \end{matrix} \\ {{- 0.4175} + {0.0837}} \end{matrix}$ ${V\; 1\left( {:{,{:{,52}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {{- 0.1140} + {0.7536}} \\ {{- 0.3076} - {0.0056}} \end{matrix} \\ {{- 0.1690} + {0.0022}} \end{matrix}$

${V\; 1\left( {:{,{:{,53}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {0 - {0.5000}} \end{matrix} \\ {0.3536 + {0.3536}} \end{matrix}$ ${V\; 1\left( {:{,{:{,54}}}} \right)} = \begin{matrix} \begin{matrix} {0.3396 + {0.1614}} \\ {{- 0.3400} - {0.3060}} \\ {0.3493 - {0.5641}} \end{matrix} \\ {{- 0.3060} + {0.3400}} \end{matrix}$ ${V\; 1\left( {:{,{:{,55}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {{- 0.1690} + {0.0022}} \\ {0.3076 + {0.0056}} \end{matrix} \\ {{- 0.1140} + {0.7536}} \end{matrix}$ ${V\; 1\left( {:{,{:{,56}}}} \right)} = \begin{matrix} \begin{matrix} {{- 0.1560} + {0.4926}} \\ {{- 0.4175} + {0.0837}} \\ {0.4597 - {0.3989}} \end{matrix} \\ {0.0837 + {0.4175}} \end{matrix}$

${V\; 1\left( {:{,{:{,57}}}} \right)} = \begin{matrix} \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0 + {0.5000}} \end{matrix} \\ {0.3536 - {0.3536}} \end{matrix}$ ${V\; 1\left( {:{,{:{,58}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {0.1140 - {0.7536}} \\ {{- 0.3076} - {0.0056}} \end{matrix} \\ {0.1690 - {0.0022}} \end{matrix}$ ${V\; 1\left( {:{,{:{,59}}}} \right)} = \begin{matrix} \begin{matrix} {0.3396 + {0.1614}} \\ {0.3060 - {0.3400}} \\ {{- 0.3493} + {0.5641}} \end{matrix} \\ {0.3400 + {0.3060}} \end{matrix}$ ${V\; 1\left( {:{,{:{,60}}}} \right)} = \begin{matrix} \begin{matrix} {0.3841 + {0.3851}} \\ {{- 0.0022} - {0.1690}} \\ {{- 0.3076} - {0.0056}} \end{matrix} \\ {0.7536 + {0.1140}} \end{matrix}$

${V\; 1\left( {:{,{:{,61}}}} \right)} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {0. - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 1\left( {:{,{:{,62}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} \\ {0.4175 - {0.0837}} \\ {0.4597 - {0.3989}} \\ {{- 0.0837} - {0.4175}} \end{matrix}$ ${V\; 1\left( {:{,{:{,63}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} \\ {0.7536 + {0.1140}} \\ {0.3076 + {0.0056}} \\ {{- 0.0022} - {0.1690}} \end{matrix}$ ${V\; 1\left( {:{,{:{,64}}}} \right)} = \begin{matrix} {0.3396 + {0.1614}} \\ {0.3400 + {0.3060}} \\ {0.3493 - {0.5641}} \\ {0.3060 - {0.3400}} \end{matrix}$

Final Rank 2 Codebook:

${V\; 2\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,2}}}} \right)} = \begin{matrix} {0.3260 + {0.6774}} & {0.4688 + {0.2170}} \\ {0.3254 + {0.1709}} & {{- 0.2948} - {0.0700}} \\ {0.3254 + {0.1709}} & {{- 0.4598} - {0.2965}} \\ {{- 0.0250} + {0.4051}} & {{- 0.5846} - {0.0154}} \end{matrix}$ ${V\; 2\left( {:{,{:{,3}}}} \right)} = \begin{matrix} {0.1499 + {0.0347}} & {{- 0.4377} + {0.4498}} \\ {0.5009 + {0.3071}} & {{- 0.1173} + {0.2079}} \\ {0.1505 + {0.5412}} & {{- 0.2098} + {0.2390}} \\ {0.1505 + {0.5412}} & {0.6133 - {0.26821}} \end{matrix}$ ${V\; 2\left( {:{,{:{,4}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {0.4585 - {0.3521}} \\ {0.1311 + {0.6387}} & {{- 0.5369} + {0.2492}} \\ {0.2473 + {0.0541}} & {{- 0.1391} + {0.3154}} \\ {0.4815 + {0.4045}} & {0.2872 - {0.3378}} \end{matrix}$

${V\; 2\left( {:{,{:{,5}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,6}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.7326} - {0.3303}} \\ {{- 0.1311} - {0.6387}} & {{- 0.1521} + {0.1245}} \\ {0.2473 + {0.0541}} & {0.0048 - {0.3893}} \\ {{- 0.4815} - {0.4045}} & {{- 0.3093} - {0.2615}} \end{matrix}$ ${V\; 2\left( {:{,{:{,7}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.4259} - {06865}} \\ {0.0056 - {0.3076}} & {0.0000 - {0.0163}} \\ {0.2285 + {0.6580}} & {{- 0.0029} + {0.5625}} \\ {{- 0.3448} - {0.0735}} & {{- 0.1664} + {0.0538}} \end{matrix}$ ${V\; 2\left( {:{,{:{,8}}}} \right)} = \begin{matrix} {0.3260 + {0.6774}} & {{- 0.2522} + {0.2830}} \\ {{- 0.3254} - {0.1709}} & {{- 0.2511} + {0.3562}} \\ {0.3254 + {0.1709}} & {{- 0.6354} - {0.2810}} \\ {0.0250 - {0.4051}} & {{- 0.2798} - {0.3246}} \end{matrix}$

${V\; 2\left( {:{,{:{,9}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} & {0.4517 - {0.1172}} \\ {{- 0.2473} - {0.0541}} & {{- 0.3568} + {0.3992}} \\ {0.1311 + {0.6387}} & {0.0918 + {0.4242}} \\ {0.4815 + {0.4045}} & {{- 0.3837} - {0.3999}} \end{matrix}$ ${V\; 2\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {{- 0.0337} - {0.6193}} & {0.3983 + {0.5597}} \\ {{- 0.4621} - {0.5019}} & {{- 0.5583} + {0.0724}} \\ {0.2280 + {0.1515}} & {0.2543 + {0.2747}} \\ {0.2280 + {0.1515}} & {{- 0.0069} + {0.2664}} \end{matrix}$ ${V\; 2\left( {:{,{:{,12}}}} \right)} = \begin{matrix} {{- 0.4422} - {0.0928}} & {0.3609 + {0.0743}} \\ {{- 0.2479} - {0.5606}} & {{- 0.0447} - {0.2624}} \\ {0.2479 + {0.5606}} & {0.2646 - {02157}} \\ {0.0137 + {0.2102}} & {{- 0.7438} + {0.3516}} \end{matrix}$

${V\; 2\left( {:{,{:{,13}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,14}}}} \right)} = \begin{matrix} {0.4422 - {0.0928}} & {{- 0.4197} + {0.4817}} \\ {0.2479 + {0.5606}} & {{- 0.2858} - {0.0562}} \\ {0.2479 + {0.5606}} & {{- 0.0712} + {0.2154}} \\ {{- 0.0137} - {0.2102}} & {0.1024 + {0.6671}} \end{matrix}$ ${V\; 2\left( {:{,{:{,15}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} & {0.3849 + {0.1986}} \\ {{- 0.0056} + {0.3076}} & {{- 0.1824} - {0.0094}} \\ {0.3448 + {0.0735}} & {0.2254 + {0.6795}} \\ {{- 0.2285} - {0.6580}} & {{- 0.5154} + {0.0292}} \end{matrix}$ ${V\; 2\left( {:{,{:{,16}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} & {0.6638 - {0.4917}} \\ {0.2473 + {0.0541}} & {{- 0.1566} - {0.1262}} \\ {0.1311 + {0.6387}} & {{- 0.0029} - {0.3366}} \\ {{- 0.4815} - {0.4045}} & {{- 0.3978} + {0.0753}} \end{matrix}$

${V\; 2\left( {:{,{:{,17}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ {- 0.5000} & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,18}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.2420 - {0.2055}} \\ {{- 0.3076} - {0.0056}} & {{- 0.5655} - {0.6754}} \\ {0.3448 + {0.0735}} & {0.0981 + {0.0467}} \\ {{- 0.6580} + {0.2285}} & {0.3298 - {0.0517}} \end{matrix}$ ${V\; 2\left( {:{,{:{,19}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.2841} - {0.7969}} \\ {{- 0.0541} + {0.2473}} & {0.0227 + {0.2058}} \\ {0.1311 + {0.6387}} & {0.2505 + {0.3816}} \\ {{- 0.4045} + {0.4815}} & {{- 0.0898} - {0.1576}} \end{matrix}$ ${V\; 2\left( {:{,{:{,20}}}} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {0.3344 - {0.1640}} \\ {{- 0.5019} + {0.4621}} & {{- 0.1475} + {0.2641}} \\ {0.2280 + {0.1515}} & {{- 0.1383} + {0.3182}} \\ {{- 0.1515} + {0.2280}} & {{- 0.3105} - {0.7437}} \end{matrix}$

${V\; 2\left( {:{,{:{,21}}}} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,22}}}} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {0.4332 + {0.1612}} \\ {0.5019 - {0.4621}} & {{- 0.2499} + {0.4647}} \\ {0.2280 + {0.1515}} & {0.1294 + {0.5446}} \\ {0.1515 - {0.2280}} & {0.3535 - {0.2640}} \end{matrix}$ ${V\; 2\left( {:{,{:{,23}}}} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {0.5948 - {0.2086}} \\ {0.5606 - {0.2479}} & {{- 0.6101} + {0.1811}} \\ {0.2479 + {0.5606}} & {{- 0.1244} + {0.3625}} \\ {0.2102 - {0.0137}} & {{- 0.1261} + {0.1870}} \end{matrix}$ ${V\; 2\left( {:{,{:{,24}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.7276} + {0.3231}} \\ {0.3076 + {0.0056}} & {0.0623 - {0.3109}} \\ {0.3448 + {0.0735}} & {{- 0.0316} - {0.3607}} \\ {0.6580 - {0.2285}} & {0.1496 - {0.3350}} \end{matrix}$

${V\; 2\left( {:{,{:{,25}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} \end{matrix}$ ${V\; 2\left( {:{,{:{,26}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} & {0.2650 + {0.1831}} \\ {0.3076 + {0.0056}} & {0.0741 + {0.1142}} \\ {0.2285 + {0.5680}} & {0.3746 + {0.4660}} \\ {{- 0.0735} + {0.3448}} & {{- 0.0601} - {0.7188}} \end{matrix}$ ${V\; 2\left( {:{,{:{,27}}}} \right)} = \begin{matrix} {{- 0.3260} - {0.6774}} & {0.4170 + {0.2673}} \\ {0.1709 - {0.3254}} & {{- 0.0605} + {0.0504}} \\ {0.3254 + {0.1709}} & {0.3145 + {0.3209}} \\ {{- 0.4051} - {0.0250}} & {{- 0.4946} + {0.5495}} \end{matrix}$ ${V\; 2\left( {:{,{:{,28}}}} \right)} = \begin{matrix} {{- 0.1499} - {0.0347}} & {{- 0.1568} + {0.5108}} \\ {0.3071 - {0.5009}} & {{- 0.2476} + {0.2774}} \\ {0.1505 + {0.5412}} & {0.1736 - {0.2279}} \\ {{- 0.5412} + {0.1505}} & {{- 0.6443} - {0.2811}} \end{matrix}$

${V\; 2\left( {:{,{:{,29}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 2\left( {:{,{:{,30}}}} \right)} = \begin{matrix} {{- 0.1499} - {0.0347}} & {{- 0.4915} + {0.5944}} \\ {{- 0.3071} + {0.5009}} & {{- 0.1046} - {0.3964}} \\ {0.1505 + {0.5412}} & {0.2288 + {0.3710}} \\ {0.5412 - {0.1505}} & {{- 0.2089} + {0.0583}} \end{matrix}$ ${V\; 2\left( {:{,{:{,31}}}} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} & {0.0849 - {0.1823}} \\ {{- 0.6387} + {0.1311}} & {{- 0.1210} - {0.0233}} \\ {0.2473 + {0.0541}} & {0.4127 - {0.7969}} \\ {0.4045 - {0.4815}} & {{- 0.0136} + {0.3727}} \end{matrix}$ ${V\; 2\left( {:{,{:{,32}}}} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} & {0.4828 - {0.2545}} \\ {{- 0.3076} - {0.0056}} & {{- 0.1939} + {0.1069}} \\ {0.2285 + {0.6580}} & {0.3116 + {0.2702}} \\ {0.0735 - {0.3448}} & {{- 0.4258} + {0.5492}} \end{matrix}$

${V\; 2\left( {:{,{:{,33}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ ${V\; 2\left( {:{,{:{,34}}}} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {{- 0.3283} - {0.4339}} \\ {0.2280 + {0.1515}} & {{- 0.3886} - {0.2987}} \\ {0.2280 + {0.1515}} & {0.2457 - {0.2337}} \\ {{- 0.4621} - {0.5019}} & {{- 0.4861} - {0.3354}} \end{matrix}$ ${V\; 2\left( {:{,{:{,35}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.0348} + {0.4484}} \\ {0.4815 + {0.4045}} & {{- 0.4400} + {0.2694}} \\ {0.1311 + {0.6387}} & {0.3429 - {0.0049}} \\ {{- 0.2473} - {0.0541}} & {0.1991 + {0.06118}} \end{matrix}$ ${V\; 2\left( {:{,{:{,36}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.2360} - {0.2973}} \\ {0.2285 + {0.6580}} & {0.0315 - {0.2366}} \\ {0.3448 + {0.0735}} & {0.7285 + {0.0791}} \\ {0.0056 - {0.3076}} & {0.4097 - {0.3067}} \end{matrix}$

${V\; 2\left( {:{,{:{,37}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 2\left( {:{,{:{,38}}}} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {{- 0.5245} - {0.3774}} \\ {{- 0.2102} + {0.0137}} & {{- 0.0522} + {0.4805}} \\ {{- 0.2479} - {0.5606}} & {0.1023 + {0.0593}} \\ {{- 0.5606} + {0.2479}} & {{- 0.5756} - {0.0591}} \end{matrix}$ ${V\; 2\left( {:{,{:{,39}}}} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {0.1953 - {0.3057}} \\ {{- 0.1515} + {0.2280}} & {{- 0.2685} - {0.1895}} \\ {{- 0.2280} - {0.1515}} & {{- 0.7779} - {0.3721}} \\ {{- 0.5019} + {0.4621}} & {{- 0.0216} - {0.1283}} \end{matrix}$ ${V\; 2\left( {:{,{:{,40}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.6035} - {0.3342}} \\ {{- 0.4045} + {0.4815}} & {{- 0.1428} + {0.4838}} \\ {{- 0.1311} - {0.6387}} & {{- 0.1927} + {0.1626}} \\ {{- 0.0541} + {0.2473}} & {{- 0.3638} - {0.2714}} \end{matrix}$

${V\; 2\left( {:{,{:{,41}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 - {0.5000}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} \end{matrix}$ ${V\; 2\left( {:{,{:{,42}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.1371 + {0.7838}} \\ {{- 0.2285} - {0.6580}} & {{- 0.0956} + {0.3657}} \\ {0.3448 + {0.0735}} & {{- 0.1099} - {0.0181}} \\ {{- 0.0056} + {0.3076}} & {0.3423 - {0.3074}} \end{matrix}$ ${V\; 2\left( {:{,{:{,43}}}} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {{- 0.0860} - {0.2241}} \\ {{- 0.0137} - {0.2102}} & {{- 0.6015} + {0.2527}} \\ {0.2479 + {0.5606}} & {{- 0.4157} - {0.2206}} \\ {0.2479 + {0.5606}} & {0.1543 + {0.5211}} \end{matrix}$ ${V\; 2\left( {:{,{:{,44}}}} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {{- 0.1081} + {0.4460}} \\ {{- 0.2280} - {0.1515}} & {{- 0.3567} - {0.5167}} \\ {0.2280 + {0.1515}} & {{- 0.2354} - {0.1914}} \\ {0.4621 + {0.5019}} & {{- 0.1985} - {0.5136}} \end{matrix}$

${V\; 2\left( {:{,{:{,45}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ 0.5000 & {0 + {0.5000}} \\ 0.5000 & 0.5000 \\ 0.5000 & {0 - {0.5000}} \end{matrix}$ ${V\; 2\left( {:{,{:{,46}}}} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.5060} + {0.1443}} \\ {0.4045 - {0.4815}} & {{- 0.4682} - {0.5571}} \\ {{- 0.1313} - {0.6387}} & {0.4345 + {0.0089}} \\ {0.0541 - {0.2473}} & {{- 0.0309} + {0.0618}} \end{matrix}$ ${V\; 2\left( {:{,{:{,47}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.0427 + {0.4037}} \\ {0.6580 - {0.2285}} & {{- 0.3221} + {0.3130}} \\ {{- 0.3448} - {0.0735}} & {{- 0.0724} + {0.1313}} \\ {0.3076 + {0.0056}} & {0.3259 - {0.7104}} \end{matrix}$ ${V\; 2\left( {:{,{:{,48}}}} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {{- 0.1512} + {0.0220}} \\ {0.2102 - {0.0137}} & {{- 0.5297} - {0.2125}} \\ {{- 0.2479} - {0.5606}} & {0.3488 + {0.2574}} \\ {0.5606 - {0.2479}} & {0.5345 - {0.4211}} \end{matrix}$

${V\; 2\left( {:{,{:{,49}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,50}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.3899 + {0.5099}} \\ {0.0022 + {0.1690}} & {{- 0.2642} - {0.2730}} \\ {{- 0.3076} - {0.0056}} & {0.0634 + {0.5454}} \\ {{- 0.7536} - {0.1140}} & {0.3743 - {0.0464}} \end{matrix}$ ${V\; 2\left( {:{,{:{,51}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} & {0.1996 + {0.3349}} \\ {0.0837 + {0.4175}} & {0.1647 + {0.4478}} \\ {{- 0.4597} + {0.3989}} & {0.0029 - {0.0825}} \\ {{- 0.4175} + {0.0837}} & {0.6245 - {0.4727}} \end{matrix}$ ${V\; 2\left( {:{,{:{,52}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.4945 - {0.2660}} \\ {{- 0.1140} + {0.7536}} & {{- 0.4531} - {0.0810}} \\ {{- 0.3076} - {0.0056}} & {0.2295 + {0.4493}} \\ {{- 0.1690} + {0.0022}} & {0.0234 - {0.4666}} \end{matrix}$

${V\; 2\left( {:{,{:{,53}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & 0.5000 \\ 0.5000 & 0.5000 \\ {0 + {0.5000}} & {- 0.5000} \end{matrix}$ ${V\; 2\left( {:{,{:{,54}}}} \right)} = \begin{matrix} {0.3396 + {0.1614}} & {{- 0.0891} - {0.1737}} \\ {{- 0.3400} - {0.3060}} & {{- 0.2160} + {0.0403}} \\ {0.3493 - {0.5641}} & {0.3530 + {0.5446}} \\ {{- 0.3060} + {0.3400}} & {0.1657 + {0.6818}} \end{matrix}$ ${V\; 2\left( {:{,{:{,55}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.3162 - {0.1330}} \\ {{- 0.1690} + {0.0022}} & {{- 0.2451} + {0.4651}} \\ {0.3076 + {0.0056}} & {0.1535 - {0.5047}} \\ {{- 0.1140} + {0.7536}} & {{- 0.4972} - {0.2836}} \end{matrix}$ ${V\; 2\left( {:{,{:{,56}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} & {0.1157 + {0.3948}} \\ {{- 0.4175} + {0.0837}} & {0.2814 - {0.3175}} \\ {0.4597 - {0.3989}} & {0.4393 + {0.0781}} \\ {0.0837 + {0.4175}} & {0.3704 - {0.5609}} \end{matrix}$

${V\; 2\left( {:{,{:{,57}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} \\ 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 2\left( {:{,{:{,58}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.6183} - {0.3893}} \\ {0.1140 - {0.7536}} & {0.0439 - {0.4301}} \\ {{- 0.3076} - {0.0056}} & {{- 0.3231} + {0.0445}} \\ {0.1690 - {0.0022}} & {{- 0.2461} - {0.3351}} \end{matrix}$ ${V\; 2\left( {:{,{:{,59}}}} \right)} = \begin{matrix} {0.3396 + {0.1614}} & {0.3566 - {0.2418}} \\ {0.3060 - {0.3400}} & {{- 0.1924} + {0.0246}} \\ {{- 0.3493} + {0.5641}} & {{- 0.5527} + {0.1626}} \\ {0.3400 + {0.3060}} & {{- 0.4030} - {0.5315}} \end{matrix}$ ${V\; 2\left( {:{,{:{,60}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.0072 + {0.6080}} \\ {{- 0.0022} - {0.1690}} & {{- 0.1531} - {0.3405}} \\ {{- 0.3076} - {0.0056}} & {0.4820 + {0.4643}} \\ {0.7536 + {0.1140}} & {{- 0.1738} - {0.1132}} \end{matrix}$

${V\; 2\left( {:{,{:{,61}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {03536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 2\left( {:{,{:{,62}}}} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} & {{- 0.4387} - {0.0343}} \\ {0.4175 - {0.0837}} & {0.5062 - {0.5972}} \\ {0.4597 - {0.3989}} & {{- 0.2611} + {0.2853}} \\ {{- 0.0837} - {0.4175}} & {{- 0.0591} + {0.2011}} \end{matrix}$ ${V\; 2\left( {:{,{:{,63}}}} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.0757} - {0.5813}} \\ {0.7536 + {0.1140}} & {0.1019 + {0.2454}} \\ {0.3076 + {0.0056}} & {0.5989 + {0.2606}} \\ {{- 0.0022} - {0.1690}} & {{- 0.3284} + {0.2264}} \end{matrix}$ ${V\; 2\left( {:{,{:{,64}}}} \right)} = \begin{matrix} {0.3396 + {0.1614}} & {0.2711 - {0.7070}} \\ {0.3400 + {0.3060}} & {{- 0.4123} - {0.1234}} \\ {0.3493 - {0.5641}} & {0.4714 - {0.0224}} \\ {0.3060 - {0.3400}} & {{- 0.0617} - {0.1224}} \end{matrix}$

Final Rank 3 Codebook:

${V\; 3\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 \end{matrix}$ ${V\; 3\left( {:{,{:{,2}}}} \right)} = \begin{matrix} {0.4688 + {0.2170}} & {0.3008 + {0.0981}} & {0.1395 - {0.2201}} \\ {{- 0.2948} - {0.0700}} & {{- 0.1167} + {0.7005}} & {{- 0.4967} + {0.1489}} \\ {{- 0.4598} - {0.2965}} & {{- 0.1297} + {0.0716}} & {0.7351 + {0.0569}} \\ {{- 0.5846} - {0.0154}} & {0.3317 - {0.5135}} & {{- 0.3430} - {0.0437}} \end{matrix}$ ${V\; 3\left( {:{,{:{,3}}}} \right)} = \begin{matrix} {{- 0.4377} + {0.4498}} & {0.2079 + {0.3012}} & {{- 0.4172} - {0.5239}} \\ {{- 0.1173} + {0.2079}} & {0.4297 + {0.2345}} & {0.4687 + {0.3722}} \\ {{- 0.2098} + {0.2390}} & {{- 0.2835} - {0.7076}} & {{- 0.0338} - {0.0330}} \\ {0.6133 - {0.2682}} & {0.1669 + {0.1324}} & {{- 0.1311} - {0.4169}} \end{matrix}$ ${V\; 3\left( {:{,{:{,4}}}} \right)} = \begin{matrix} {0.4585 - {0.3521}} & {0.4797 - {0.5371}} & {0.1760 - {0.0285}} \\ {{- 0.5369} + {0.2492}} & {0.2256 + {0.0214}} & {0.0252 - {0.4154}} \\ {{- 0.1391} + {0.3154}} & {{- 0.1147} - {0.1411}} & {0.7276 + {0.5046}} \\ {0.2872 - {0.3378}} & {{- 0.2334} + {0.5852}} & {0.0121 + {0.1041}} \end{matrix}$

${V\; 3\left( {:{,{:{,5}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,6}}}} \right)} = \begin{matrix} {{- 0.7326} - {0.3303}} & {0.1180 - {0.0570}} & {0.3116 - {0.3531}} \\ {{- 0.1521} + {0.1245}} & {{- 0.3996} - {0.1865}} & {0.5845 - {0.0141}} \\ {0.0048 - {0.3893}} & {{- 0.8009} - {0.0266}} & {{- 0.3599} - {0.1123}} \\ {{- 0.3093} - {0.2615}} & {0.1996 - {0.3262}} & {{- 0.5417} + {0.0279}} \end{matrix}$ ${V\; 3\left( {:{,{:{,7}}}} \right)} = \begin{matrix} {{- 0.4259} - {0.6865}} & {0.1888 - {0.1148}} & {0.0505 - {0.0112}} \\ {0.0000 - {0.0163}} & {0.5849 + {0.0396}} & {0.1477 + {0.7346}} \\ {{- 0.0029} + {0.5625}} & {0.2943 + {0.2161}} & {0.2547 - {0.0108}} \\ {{- 0.1664} + {0.0538}} & {0.6828 - {0.0892}} & {{- 0.3558} - {0.4943}} \end{matrix}$ ${V\; 3\left( {:{,{:{,8}}}} \right)} = \begin{matrix} {{- 0.2522} + {0.2830}} & {0.0842 - {0.4310}} & {0.0964 + {0.2984}} \\ {{- 0.2511} + {0.3562}} & {{- 0.5001} - {0.4577}} & {{- 0.4236} - {0.1898}} \\ {{- 0.6354} - {0.2810}} & {{- 0.3335} + {0.3466}} & {{- 0.1484} - {0.2414}} \\ {{- 0.2798} - {0.3246}} & {{- 0.1512} - {0.1513}} & {0.0306 + {0.7778}} \end{matrix}$

${V\; 3\left( {:{,{:{,9}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,10}}}} \right)} = \begin{matrix} {0.4517 - {0.1172}} & {{- 0.0439} + {0.5591}} & {0.3594 - {0.4725}} \\ {{- 0.3568} + {0.3992}} & {{- 0.4696} - {0.0932}} & {{- 0.2384} - {0.6026}} \\ {0.0918 + {0.4242}} & {0.4085 - {0.0481}} & {0.3325 - {0.3269}} \\ {{- 0.3837} - {0.3999}} & {{- 0.3005} + {0.4437}} & {{- 0.0155} - {0.1000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,11}}}} \right)} = \begin{matrix} {0.3983 + {0.5597}} & {{- 0.1922} - {0.0358}} & {{- 0.2096} + {0.2475}} \\ {{- 0.5583} + {0.0724}} & {0.2420 - {0.0149}} & {0.3931 - {0.0653}} \\ {0.2543 + {0.2747}} & {0.7756 - {0.0378}} & {0.3367 + {0.2619}} \\ {{- 0.0069} + {0.2664}} & {{- 0.5242} + {0.1589}} & {0.7417 + {0.0629}} \end{matrix}$ ${V\; 3\left( {:{,{:{,12}}}} \right)} = \begin{matrix} {0.3609 + {0.0743}} & {{- 0.6309} - {0.1183}} & {0.2159 - {0.4488}} \\ {{- 0.0447} - {0.2624}} & {{- 0.3156} + {0.2879}} & {{- 0.0988} + {0.6010}} \\ {0.2646 - {0.2157}} & {{- 0.3493} - {0.1821}} & {0.3113 + {0.5056}} \\ {{- 0.7438} + {0.3516}} & {{- 0.4975} - {0.0539}} & {{- 0.1643} + {0.0375}} \end{matrix}$

${V\; 3\left( {:{,{:{,13}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,14}}}} \right)} = \begin{matrix} {{- 0.4197} + {0.4817}} & {0.0820 + {0.2932}} & {0.5394 + {0.0634}} \\ {{- 0.2858} - {0.0562}} & {0.0717 - {0.5527}} & {0.3834 + {0.2860}} \\ {{- 0.0712} + {0.2154}} & {{- 0.2709} + {0.6053}} & {{- 0.2652} + {0.2505}} \\ {0.1024 + {0.6671}} & {{- 0.0386} - {0.3943}} & {{- 0.3941} + {0.4333}} \end{matrix}$ ${V\; 3\left( {:{,{:{,15}}}} \right)} = \begin{matrix} {0.3849 + {0.1986}} & {{- 0.5420} - {0.2820}} & {0.3395 + {0.1672}} \\ {{- 0.1824} - {0.0094}} & {0.1989 + {0.1166}} & {0.6694 + {0.6089}} \\ {0.2254 + {0.6795}} & {{- 0.1979} + {0.5597}} & {{- 0.0690} + {0.0775}} \\ {{- 0.5154} + {0.0292}} & {0.0915 + {0.4612}} & {0.1367 - {0.0921}} \end{matrix}$ ${V\; 3\left( {:{,{:{,16}}}} \right)} = \begin{matrix} {0.6638 - {0.4917}} & {{- 0.0134} - {0.1109}} & {0.3780 + {0.2166}} \\ {{- 0.1566} - {0.1262}} & {0.0722 - {0.9308}} & {{- 0.1132} + {0.1055}} \\ {{- 0.0029} - {0.3366}} & {{- 0.1758} + {0.1916}} & {{- 0.1479} + {0.6100}} \\ {{- 0.3978} + {0.0753}} & {0.2200 - {0.0040}} & {{- 0.0224} + {0.6259}} \end{matrix}$

${V\; 3\left( {:{,{:{,17}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,18}}}} \right)} = \begin{matrix} {0.2420 - {0.2055}} & {{- 0.1388} + {0.6904}} & {0.3273 + {0.0199}} \\ {{- 0.5655} - {0.6754}} & {{- 0.2104} + {0.2029}} & {{- 0.1634} + {0.1312}} \\ {0.0981 + {0.0467}} & {0.1060 + {0.1310}} & {{- 0.7933} + {0.4542}} \\ {0.3298 - {0.0517}} & {0.5748 + {0.2447}} & {{- 0.0516} + {0.1018}} \end{matrix}$ ${V\; 3\left( {:{,{:{,19}}}} \right)} = \begin{matrix} {{- 0.2841} - {0.7969}} & {0.3713 - {0.0745}} & {0.0639 + {0.1460}} \\ {0.0227 + {0.2058}} & {0.5630 - {0.0230}} & {{- 0.0999} - {0.7520}} \\ {0.2505 + {0.3816}} & {0.1527 - {0.2390}} & {0.3715 + {0.3849}} \\ {{- 0.0898} - {0.1576}} & {{- 0.6437} - {0.2105}} & {0.0144 - {0.3358}} \end{matrix}$ ${V\; 3\left( {:{,{:{,20}}}} \right)} = \begin{matrix} {0.3344 - {0.1640}} & {{- 0.2485} + {0.4791}} & {0.4000 + {0.1590}} \\ {{- 0.1475} + {0.2641}} & {0.5893 + {0.1330}} & {{- 0.1383} - {0.2427}} \\ {{- 0.1383} + {0.3182}} & {{- 0.4811} + {0.3240}} & {{- 0.1561} - {0.4493}} \\ {{- 0.3105} - {0.7437}} & {{- 0.0811} - {0.0251}} & {{- 0.4817} + {0.1909}} \end{matrix}$

${V\; 3\left( {:{,{:{,21}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,22}}}} \right)} = \begin{matrix} {0.4332 + {0.1612}} & {{- 0.4077} - {0.0486}} & {0.4750 + {0.0865}} \\ {{- 0.2499} + {0.4647}} & {{- 0.3727} - {0.1094}} & {0.2747 + {0.1727}} \\ {0.1294 + {0.5446}} & {0.6827 - {0.1403}} & {0.1161 - {0.3354}} \\ {0.3535 - {0.2640}} & {0.4328 + {0.0863}} & {0.3135 + {0.6613}} \end{matrix}$ ${V\; 3\left( {:{,{:{,23}}}} \right)} = \begin{matrix} {0.5948 - {0.2086}} & {{- 0.6155} + {0.0029}} & {0.0246 + {0.1384}} \\ {{- 0.6101} + {0.1811}} & {{- 0.2226} - {0.2992}} & {0.0747 + {0.2731}} \\ {{- 0.1244} + {0.3625}} & {0.0659 + {0.6219}} & {0.2542 + {0.1473}} \\ {{- 0.1261`} + {0.1870}} & {{- 0.1866} + {0.2370}} & {{- 0.7184} - {0.5456}} \end{matrix}$ ${V\; 3\left( {:{,{:{,24}}}} \right)} = \begin{matrix} {{- 0.7276} + {0.3231}} & {{- 0.0746} + {0.0002}} & {0.1968 + {0.1617}} \\ {0.0623 - {0.3109}} & {0.7114 + {0.3826}} & {0.3902 + {0.0102}} \\ {{- 0.0316} - {0.3607}} & {{- 0.4356} + {0.1774}} & {0.0838 - {0.7186}} \\ {0.1496 - {0.3350}} & {{- 0.1859} - {0.2936}} & {{- 0.1918} + {0.4718}} \end{matrix}$

${V\; 3\left( {:{,{:{,25}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,26}}}} \right)} = \begin{matrix} {0.2560 + {0.1831}} & {0.2046 + {0.4515}} & {0.2200 + {0.5535}} \\ {0.0741 + {0.1142}} & {{- 0.2010} + {0.6413}} & {0.4491 - {0.4832}} \\ {0.3746 + {0.4660}} & {0.2087 + {0.1370}} & {{- 0.2423} + {0.1904}} \\ {{- 0.0601} - {0.7188}} & {{- 0.1085} + {0.4782}} & {{- 0.2656} + {0.2110}} \end{matrix}$ ${V\; 3\left( {:{,{:{,27}}}} \right)} = \begin{matrix} {0.4170 + {0.2673}} & {0.0729 - {0.3667}} & {{- 0.1258} + {0.1841}} \\ {{- 0.0605} + {0.0504}} & {{- 0.2440} + {0.1836}} & {0.8489 + {0.2116}} \\ {0.3145 + {0.3209}} & {{- 0.6599} - {0.3142}} & {{- 0.0292} - {0.3579}} \\ {{- 0.4946} + {0.5495}} & {{- 0.3770} + {0.3011}} & {{- 0.2007} + {0.1251}} \end{matrix}$ ${V\; 3\left( {:{,{:{,28}}}} \right)} = \begin{matrix} {{- 0.1568} + {0.5108}} & {{- 0.6900} + {0.1066}} & {0.4419 + {0.0901}} \\ {{- 0.2476} + {0.2774}} & {0.3823 - {0.1521}} & {0.3815 - {0.4492}} \\ {0.1736 - {0.2279}} & {{- 0.1974} - {0.4946}} & {0.3241 - {0.4623}} \\ {{- 0.6443} - {0.2811}} & {0.0584 + {0.2373}} & {0.0318 - {0.3599}} \end{matrix}$

${V\; 3\left( {:{,{:{,29}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 - {0.5000}} & {0. - {5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,30}}}} \right)} = \begin{matrix} {{- 0.4915} + {0.5944}} & {0.0644 - {0.0145}} & {{- 0.1689} - {0.5903}} \\ {{- 0.1046} - {0.3964}} & {{- 0.5642} + {0.0031}} & {{- 0.3416} - {0.2274}} \\ {0.2288 + {0.3710}} & {{- 0.2561} - {0.0135}} & {0.6471 - {0.0998}} \\ {{- 0.2089} + {0.0583}} & {{- 0.4070} - {0.6677}} & {{- 0.1176} + {0.1097}} \end{matrix}$ ${V\; 3\left( {:{,{:{,31}}}} \right)} = \begin{matrix} {0.0849 - {0.1823}} & {0.5488 + {0.2367}} & {{- 0.3534} - {0.6018}} \\ {{- 0.1210} - {0.0233}} & {{- 0.2347} - {0.3498}} & {0.3248 - {0.5261}} \\ {0.4127 - {0.7969}} & {{- 0.2180} - {0.2794}} & {{- 0.0107} - {0.0700}} \\ {{- 0.0136} + {0.3727}} & {{- 0.3369} - {0.4757}} & {{- 0.2087} - {0.2866}} \end{matrix}$ ${V\; 3\left( {:{,{:{,32}}}} \right)} = \begin{matrix} {0.4828 - {0.2545}} & {0.2119 + {0.3871}} & {{- 0.3824} - {0.2556}} \\ {{- 0.1939} + {0.1069}} & {{- 0.3495} + {0.6691}} & {0.1365 + {0.5175}} \\ {0.3116 + {0.2702}} & {{- 0.1220} + {0.2056}} & {{- 0.5361} - {0.0030}} \\ {{- 0.4258} + {0.5492}} & {0.4208 + {0.0338}} & {{- 0.4460} + {0.1251}} \end{matrix}$

${V\; 3\left( {:{,{:{,33}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,34}}}} \right)} = \begin{matrix} {{- 0.3283} - {0.4339}} & {{- 0.2690} + {0.1335}} & {{- 0.4536} - {0.1528}} \\ {{- 0.3886} - {0.2987}} & {0.0982 - {0.6284}} & {0.5255 + {0.0653}} \\ {0.2457 - {0.23371}} & {{- 0.3618} + {0.5436}} & {0.6122 + {0.0946}} \\ {{- 0.4861} - {0.3354}} & {{- 0.2060} + {0.1911}} & {0.0216 + {0.3261}} \end{matrix}$ ${V\; 3\left( {:{,{:{,35}}}} \right)} = \begin{matrix} {{- 0.0348} + {0.4484}} & {{- 0.4669} - {0.6204}} & {0.2685 + {0.0858}} \\ {{- 0.4400} + {0.2694}} & {0.0511 + {0.4386}} & {{- 0.1700} + {0.3385}} \\ {0.3429 - {0.0049}} & {0.4475 - {0.0066}} & {0.1480 - {0.4849}} \\ {0.1991 + {0.6118}} & {0.0419 + {0.0073}} & {{- 0.6988} - {0.1782}} \end{matrix}$ ${V\; 3\left( {:{,{:{,36}}}} \right)} = \begin{matrix} {{- 0.2360} - {0.2973}} & {0.3316 + {0.2303}} & {0.0526 + {0.6279}} \\ {0.0315 - {0.2366}} & {{- 0.3698} - {0.2862}} & {0.3197 - {0.3701}} \\ {0.7285 + {0.0791}} & {0.5225 + {0.0720}} & {{- 0.0255} - {0.2448}} \\ {0.4097 - {0.3067}} & {{- 0.4142} + {0.4106}} & {0.5048 + {0.2198}} \end{matrix}$

${V\; 3\left( {:{,{:{,37}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,38}}}} \right)} = \begin{matrix} {{- 0.5245} - {0.3774}} & {{- 0.5623} - {0.0655}} & {0.2278 + {0.0771}} \\ {{- 0.0522} + {0.4805}} & {{- 0.1132} - {0.5413}} & {0.6451 - {0.0046}} \\ {0.1023 + {0.0593}} & {{- 0.5232} - {0.2730}} & {{- 0.4236} + {0.2873}} \\ {{- 0.5756} - {0.0591}} & {0.0088 - {0.1592}} & {{- 0.3254} - {0.3977}} \end{matrix}$ ${V\; 3\left( {:{,{:{,39}}}} \right)} = \begin{matrix} {0.1953 - {0.3057}} & {{- 0.5062} + {0.0964}} & {0.4500 + {0.1253}} \\ {{- 0.2685} - {0.1895}} & {0.6204 + {0.3044}} & {0.1623 + {0.5596}} \\ {{- 0.7779} - {0.3721}} & {{- 0.3442} - {0.2479{\iota i}}} & {{- 0.0309} + {0.0263}} \\ {{- 0.0216} - {0.1283}} & {0.2771 + {0.0109}} & {{- 0.1897} - {0.6361}} \end{matrix}$ ${V\; 3\left( {:{,{:{,40}}}} \right)} = \begin{matrix} {{- 0.6035} - {0.3342}} & {{- 0.3380} - {0.3706}} & {{- 0.0686} + {0.3905}} \\ {{- 0.1428} + {0.4838}} & {{- 0.1430} - {0.1006}} & {{- 0.3957} - {0.4036}} \\ {{- 0.1927} + {0.1626}} & {{- 0.0874} + {0.0678}} & {{- 0.6478} + {0.2818}} \\ {{- 0.3638} - {0.2714}} & {0.3209 + {0.7763}} & {{- 0.1557} - {0.0019}} \end{matrix}$

${V\; 3\left( {:{,{:{,41}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 \end{matrix}$ ${V\; 3\left( {:{,{:{,42}}}} \right)} = \begin{matrix} {0.1371 + {0.7838}} & {0.2479 + {0.0123}} & {{- 0.0008} - {0.0975}} \\ {{- 0.0956} + {0.3657}} & {0.2440 + {0.34471}} & {{- 0.4391} - {0.0271}} \\ {{- 0.1099} - {0.0181}} & {{- 0.2141} + {0.1937}} & {{- 0.2988} + {0.8311}} \\ {0.3423 - {0.3074}} & {0.2247 + {0.7913}} & {{- 0.0349} - {0.1254}} \end{matrix}$ ${V\; 3\left( {:{,{:{,43}}}} \right)} = \begin{matrix} {{- 0.0860} - {0.2241}} & {0.5338 - {0.0441}} & {{- 0.1033} + {0.6639}} \\ {{- 0.6015} + {0.2527}} & {{- 0.4372} + {0.3957}} & {{- 0.1471} + {0.4008}} \\ {{- 0.4157} - {0.2206}} & {0.0012 + {0.2819}} & {0.4969 - {0.2768}} \\ {0.1543 + {0.5211}} & {{- 0.3515} - {0.4030}} & {{- 0.0551} + {0.1997}} \end{matrix}$ ${V\; 3\left( {:{,{:{,44}}}} \right)} = \begin{matrix} {{- 0.1081} + {0.4460}} & {0.0435 - {0.3239}} & {{- 0.1491} + {0.5251}} \\ {{- 0.3567} - {0.5167}} & {{- 0.1969} - {0.3585}} & {0.3858 + {0.4634}} \\ {{- 0.2354} - {0.1914}} & {0.2532 + {0.7745}} & {0.0863 + {0.4020}} \\ {{- 0.1985} - {0.5136}} & {0.0895 - {0.2322}} & {{- 0.0884} - {0.4019}} \end{matrix}$

${V\; 3\left( {:{,{:{,45}}}} \right)} = \begin{matrix} 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 3\left( {:{,{:{,46}}}} \right)} = \begin{matrix} {{- 0.5060} + {0.1443}} & {0.5534 - {0.3836}} & {{- 0.1155} - {0.3757}} \\ {{- 0.4682} - {0.5571}} & {{- 0.1369} - {0.0572}} & {{- 0.2290} + {0.0234}} \\ {0.4345 + {0.0089}} & {0.5727 - {0.1978}} & {{- 0.1012} - {0.0932}} \\ {{- 0.0309} + {0.0618{\iota i}}} & {{- 0.2903} - {0.2707}} & {0.7177 - {0.5085}} \end{matrix}$ ${V\; 3\left( {:{,{:{,47}}}} \right)} = \begin{matrix} {0.0427 + {0.4037}} & {0.1038 + {0.3577}} & {0.5172 + {0.3648}} \\ {{- 0.3221} + {0.3130}} & {{- 0.3384} - {0.3674}} & {{- 0.2477} + {0.0474}} \\ {{- 0.0724} + {0.1313}} & {{- 0.0231} - {0.7092}} & {0.5412 + {0.2385}} \\ {0.3259 - {0.7104}} & {{- 0.3205} - {0.0747}} & {0.1184 + {0.4148}} \end{matrix}$ ${V\; 3\left( {:{,{:{,48}}}} \right)} = \begin{matrix} {{- 0.1512} + {0.0220}} & {{- 0.5578} + {0.3060}} & {{- 0.4803} + {0.4790}} \\ {{- 0.5297} - {0.2125}} & {{- 0.4245} + {0.2405}} & {0.3258 - {0.5345}} \\ {0.3488 + {0.2574}} & {{- 0.5660} - {0.2718}} & {{- 0.0021} - {0.2051}} \\ {0.5345 - {0.4211}} & {0.1552 + {0.1767}} & {{- 0.1657} - {0.2802}} \end{matrix}$

${V\; 3\left( {:{,{:{,49}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} \\ 0.5000 & {- 0.5000} & 0.5000 \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 \end{matrix}$ ${V\; 3\left( {:{,{:{,50}}}} \right)} = \begin{matrix} {0.3899 + {0.5099}} & {0.1239 + {0.3480}} & {{- 0.3212} + {0.2294}} \\ {{- 0.2642} - {0.2730}} & {{- 0.0709} + {0.7215}} & {0.4401 + {0.3283}} \\ {0.0634 + {0.5454}} & {{- 0.0400} - {0.3119}} & {0.5426 + {0.4590}} \\ {0.3743 - {0.0464}} & {0.3735 + {0.3156}} & {{- 0.1921} - {0.0291}} \end{matrix}$ ${V\; 3\left( {:{,{:{,51}}}} \right)} = \begin{matrix} {0.1996 + {0.3349}} & {0.3567 - {0.6608}} & {0.1085 + {0.0728}} \\ {0.1647 + {0.4478}} & {{- 0.0991} + {0.3844}} & {{- 0.5490} - {0.3633}} \\ {0.0029 - {0.0825}} & {{- 0.4340} + {0.1941}} & {{- 0.0351} + {0.6288}} \\ {0.6245 - {0.4727}} & {0.1497 + {0.1732}} & {0.1381 - {0.3657}} \end{matrix}$ ${V\; 3\left( {:{,{:{,52}}}} \right)} = \begin{matrix} {0.4945 - {0.2660}} & {{- 0.4949} + {0.3789}} & {{- 0.0039} - {0.0184}} \\ {{- 0.4531} - {0.0810}} & {0.2332 + {0.1714}} & {{- 0.0493} - {0.3478}} \\ {0.2295 + {0.4493}} & {0.2178 + {0.6234}} & {{- 0.4191} + {0.1977}} \\ {0.0234 - {0.4666}} & {{- 0.0755} - {0.2931}} & {{- 0.8112} + {0.0581}} \end{matrix}$

${V\; 3\left( {:{,{:{,53}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0. - {0.5000}} \\ 0.5000 & 0.5000 & {- 0.5000} \\ {- 0.5000} & 0.5000 & {0 - {0.5000}} \end{matrix}$ ${V\; 3\left( {:{,{:{,54}}}} \right)} = \begin{matrix} {{- 0.0891} - {0.1737}} & {{- 0.5874} + {0.0189}} & {0.4181 + {0.5480}} \\ {{- 0.2160} + {0.0403}} & {{- 04608} - {0.5955}} & {0.2155 - {0.3594}} \\ {0.3530 + {0.5446}} & {{- 0.2058} - {0.0338}} & {{- 0.2629} + {0.1610}} \\ {0.1657 + {0.6818}} & {0.1876 - {0.0948}} & {0.4723 + {0.1766}} \end{matrix}$ ${V\; 3\left( {:{,{:{,55}}}} \right)} = \begin{matrix} {0.3162 - {0.1330}} & {{- 0.5287} - {0.1679}} & {0.1564 + {0.5043}} \\ {{- 0.2451} + {0.4651}} & {0.2831 + {0.1482}} & {0.4827 + {0.6000}} \\ {0.1535 - {0.5047}} & {0.7325 - {0.1862}} & {0.0058 + {0.2362}} \\ {{- 0.4972} - {0.2836}} & {0.0497 + {0.1282}} & {0.1692 - {0.2094}} \end{matrix}$ ${V\; 3\left( {:{,{:{,56}}}} \right)} = \begin{matrix} {0.1157 + {0.3948}} & {0.5787 - {0.3292}} & {0.2943 - {0.1843}} \\ {0.2814 - {0.3175}} & {0.2525 - {0.2620}} & {{- 0.5641} + {0.4338}} \\ {0.4393 + {0.0781}} & {0.3512 + {0.0363}} & {0.2819 + {0.4758}} \\ {0.3704 - {0.5609}} & {0.1336 + {0.5309}} & {0.1521 - {0.2100}} \end{matrix}$

${V\; 3\left( {:{,{:{,57}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 3\left( {:{,{:{,58}}}} \right)} = \begin{matrix} {{- 0.6183} - {0.3893}} & {{- 0.3104} - {0.0769}} & {0.2262 - {0.1301}} \\ {0.0439 - {0.4301}} & {0.0424 - {0.0174}} & {0.4717 - {0.0872}} \\ {{- 0.3231} + {0.0445}} & {{- 0.0056} + {0.4980}} & {0.2818 + {0.6867}} \\ {{- 0.2461} - {0.3351}} & {0.7828 + {0.1865}} & {{- 0.3883} + {0.0121}} \end{matrix}$ ${V\; 3\left( {:{,{:{,59}}}} \right)} = \begin{matrix} {0.3566 - {0.2418}} & {0.0169 - {0.4765}} & {{- 0.5447} - {0.3859}} \\ {{- 0.1924} + {0.0246}} & {{- 0.3480} + {0.4512}} & {0.0113 - {0.6545}} \\ {{- 0.5527} + {0.1626}} & {{- 0.3128} - {0.0721}} & {{- 0.3061} - {0.1765}} \\ {{- 0.4030} - {0.5315}} & {0.5097 + {0.2917}} & {0.0150 + {0.0288}} \end{matrix}$ ${V\; 3\left( {:{,{:{,60}}}} \right)} = \begin{matrix} {0.0072 + {0.6080}} & {0.0164 + {0.4489}} & {0.2019 + {0.3032}} \\ {{- 0.1531} - {0.34051}} & {{- 0.1388} + {0.8192}} & {0.0946 - {0.3643}} \\ {0.4820 + {0.4643}} & {{- 0.2314} + {0.1231}} & {{- 0.4842} - {0.3928}} \\ {{- 0.1738} - {0.1132}} & {{- 0.1247} - {0.1538}} & {{- 0.4937} - {0.3051}} \end{matrix}$

${V\; 3\left( {:{,{:{,61}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${V\; 3\left( {:{,{:{,62}}}} \right)} = \begin{matrix} {{- 0.4387} - {0.0343}} & {0.5100 - {0.0764}} & {0.2952 + {0.4317}} \\ {0.5062 - {0.5972}} & {0.2808 - {0.2160}} & {0.1570 + {0.2361}} \\ {{- 0.2611} + {0.2853}} & {{- 0.1195} + {0.0730}} & {{- 0.1784} + {0.6546}} \\ {{- 0.0591} + {0.2011}} & {0.7066 - {0.2995}} & {{- 0.3568} - {0.2417}} \end{matrix}$ ${V\; 3\left( {:{,{:{,63}}}} \right)} = \begin{matrix} {{- 0.0757} - {0.5813}} & {0.1457 - {0.0933}} & {{- 0.5106} + {0.2643}} \\ {0.1019 + {0.2454}} & {0.3241 + {0.2186}} & {0.2092 - {0.3897}} \\ {0.5989 + {0.2606}} & {{- 0.5728} - {0.0137}} & {{- 0.1198} + {0.3689}} \\ {{- 0.3284} + {0.2264}} & {{- 0.1976} + {0.0678}} & {{- 0.5620} - {0.0868}} \end{matrix}$ ${V\; 3\left( {:{,{:{,64}}}} \right)} = \begin{matrix} {0.2711 - {0.7070}} & {{- 0.2911} + {0.2527}} & {{- 0.1768} + {0.3248}} \\ {{- 0.4123} - {0.1234}} & {0.0110 - {0.0641}} & {{- 0.4612} - {0.6234}} \\ {0.4714 - {0.0224}} & {0.3098 + {0.2417}} & {0.0189 - {0.4271}} \\ {{- 0.0617} - {0.1224}} & {{- 0.4468} - {0.7023}} & {0.2720 - {0.0718}} \end{matrix}$

Final Rank 4 Codebook:

${V\; 4\left( {:{,{:{,1}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \end{matrix}$ ${V\; 4\left( {:{,{:{,2}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 4\left( {:{,{:{,3}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ 0.5000 & {- 0.5000} & {0 - {0.5000}} & {0 + {0.5000}} \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {- 0.5000} & {0 + {0.5000}} & {0 - {0.5000}} \end{matrix}$ ${V\; 4\left( {:{,{:{,4}}}} \right)} = \begin{matrix} 0.5000 & 0.5000 & {- 0.5000} & {- 0.5000} \\ {0 + {0.5000}} & {0 - {0.5000}} & {- 0.5000} & 0.5000 \\ 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0 + {0.5000}} & {0 - {0.5000}} & 0.5000 & {- 0.5000} \end{matrix}$

${V\; 4\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0 + {0.5000}} & {- 0.5000} & {0 - {0.5000}} \\ 0.5000 & {- 0.5000} & 0.5000 & {- 0.5000} \\ {- 0.5000} & {0 + {0.5000}} & 0.5000 & {0 - {0.5000}} \end{matrix}$ ${V\; 4\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0 + {0.5000}} & {0 - {0.5000}} & {0 + {0.5000}} & {0 - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {0.3260 + {0.6774}} & {0.4688 + {0.2170}} & {0.3008 + {0.0981}} & {0.1395 - {0.2201}} \\ {0.3254 + {0.1709}} & {{- 0.2948} - {0.0700}} & {{- 0.1167} + {0.7005}} & {{- 0.4967} + {0.1489}} \\ {0.3254 + {0.1709}} & {{- 0.4598} - {0.2965}} & {{- 0.1297} + {0.0716}} & {0.7351 + {0.0569}} \\ {{- 0.0250} + {0.4051}} & {{- 0.5846} - {0.0154}} & {0.3317 - {0.5135}} & {{- 0.3430} - {0.0437}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.1499 + {0.0347}} & {{- 0.4377} + {0.4498}} & {0.2079 + {0.3012}} & {{- 0.4172} - {0.5239}} \\ {0.5009 + {0.3071}} & {{- 0.1173} + {0.2079}} & {0.4297 + {0.2345}} & {0.4687 + {0.3722}} \\ {0.1505 + {0.5412}} & {{- 0.2098} + {0.2390}} & {{- 0.2835} - {0.7076}} & {{- 0.0338} - {0.0330}} \\ {0.1505 + {0.5412}} & {0.6133 - {0.2682}} & {0.1669 + {0.1324}} & {{- 0.1311} - {0.4169}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.0918 + {0.3270`}} & {0.4585 - {0.3521}} & {0.4797 - {0.5371}} & {0.1760 - {0.0285}} \\ {0.1311 + {0.6387}} & {{- 0.5369} + {0.2492}} & {0.2256 + {0.0214}} & {0.0252 - {0.4154}} \\ {0.2473 + {0.0541}} & {{- 0.1391} + {0.3154}} & {{- 0.1147} - {0.1411}} & {0.7276 + {0.5046}} \\ {0.4815 + {0.4045}} & {0.2872 - {0.3378}} & {{- 0.2334} + {0.5852}} & {0.0121 + {0.1041}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.0918 + {0.3270`}} & {{- 0.7326} - {0.3303}} & {0.1180 - {0.0570}} & {0.3116 - {0.3531}} \\ {{- 0.1311} - {0.6387}} & {{- 0.1521} + {0.1245}} & {{- 0.3996} - {0.1865}} & {0.5845 - {0.0141}} \\ {0.2473 + {0.0541}} & {0.0048 - {0.3893}} & {{- 0.8009} - {0.0266}} & {{- 0.3599} - {0.1123}} \\ {{- 0.4815} - {0.4045}} & {{- 0.3093} - {0.2615}} & {0.1996 - {0.3262}} & {{- 0.5417} + {0.0279}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.4259} - {0.6865}} & {0.1888 - {0.1148}} & {0.0505 - {0.0112}} \\ {0.0056 - {0.3076}} & {0.0000 - {0.0163}} & {0.5849 + {0.0396}} & {0.1477 + {0.7346}} \\ {0.2285 + {0.6580}} & {{- 0.0029} + {0.5625}} & {0.2943 + {0.2161}} & {0.2547 - {0.0108}} \\ {{- 0.3448} - {0.0735}} & {{- 0.1664} + {0.0538}} & {0.6828 - {0.0892}} & {{- 0.3558} - {0.4943}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.3260 + {0.6774}} & {{- 0.2522} + {0.2830}} & {0.0842 - {0.4310}} & {0.0964 + {0.2984}} \\ {{- 0.3254} - {0.1709}} & {{- 0.2511} + {0.3562}} & {{- 0.5001} - {0.4577}} & {{- 0.4236} - {0.1898}} \\ {0.3254 + {0.1709}} & {{- 0.6354} - {0.2810}} & {{- 0.3335} + {0.4366}} & {{- 0.1484} - {0.2414}} \\ {0.0250 - {0.4051}} & {{- 0.2798} - {0.3246}} & {{- 0.1512} - {0.1513}} & {0.0306 + {0.7778}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} & {0.4517 - {0.1172}} & {{- 0.0439} + {0.5591}} & {0.3594 - {0.4725}} \\ {{- 0.2473} - {0.0541}} & {{- 0.3568} + {0.3992}} & {{- 0.4696} - {0.0932}} & {{- 0.2384} - {0.6026}} \\ {0.1311 + {0.6387}} & {0.0918 + {0.4242}} & {0.4085 - {0.0481}} & {0.3325 - {0.3269}} \\ {0.4815 + {0.4045}} & {{- 0.3837} - {0.3999}} & {{- 0.3005} + {0.4437}} & {{- 0.0155} - {0.1000}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.0337} - {0.6193`}} & {0.3983 + {0.5597}} & {{- 0.1922} - {0.0358}} & {{- 0.2096} + {0.2475}} \\ {{- 0.4621} - {0.5019}} & {{- 0.5583} + {0.0724}} & {0.2420 - {0.0149}} & {0.3931 - {0.0653}} \\ {0.2280 + {0.1515}} & {0.2543 + {0.2747}} & {0.7756 - {0.0378}} & {0.3367 + {0.2619}} \\ {0.2280 + {0.1515}} & {{- 0.0069} + {0.2664}} & {{- 0.5242} + {0.1589}} & {0.7417 + {0.0629}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {{- 0.4422} - {0.0928}} & {0.3609 + {0.0743}} & {{- 0.6309} - {0.1183}} & {0.2159 - {0.4488}} \\ {{- 0.2479} - {0.5606}} & {{- 0.0447} - {0.2624}} & {{- 0.3156} + {0.2879}} & {{- 0.0988} + {0.6010}} \\ {0.2479 + {0.5606}} & {0.2646 - {0.2157}} & {{- 0.3493} - {0.1821}} & {0.3113 + {0.5056}} \\ {0.0137 + {0.2102}} & {{- 0.7438} + {0.3516}} & {{- 0.4975} - {0.0539}} & {{- 0.1643} + {0.0375}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {{- 0.4422} - {0.0928}} & {{- 0.4197} + {0.4817}} & {0.0820 + {0.2932}} & {0.5394 + {0.0634}} \\ {0.2479 + {0.5606}} & {{- 0.2858} - {0.0562}} & {0.0717 - {0.5527}} & {0.3834 + {0.2860}} \\ {0.2479 + {0.5606}} & {{- 0.0712} + {0.2154}} & {{- 0.2709} + {0.6053}} & {{- 0.2652} + {0.2505}} \\ {{- 0.0137} - {0.2102}} & {0.1024 + {0.6671}} & {{- 0.0386} - {0.3943}} & {{- 0.3941} + {0.4333}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},17} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} & {0.3849 + {0.1986`}} & {{- 0.5420} - {0.2820}} & {0.3395 + {0.1672}} \\ {{- 0.0056} + {0.3076}} & {{- 0.1824} - {0.0094}} & {0.1989 + {0.1166}} & {0.6694 + {0.6089}} \\ {0.3448 + {0.0735}} & {0.2254 + {0.6795}} & {{- 0.1979} + {0.5597}} & {{- 0.0690} + {0.0775}} \\ {{- 0.2285} - {0.6580}} & {{- 0.5154} + {0.0292}} & {0.0915 + {0.4612}} & {0.1367 - {0.0921}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} & {0.6638 - {0.4917}} & {{- 0.0134} - {0.1109}} & {0.3780 + {0.2166}} \\ {0.2473 + {0.0541}} & {{- 0.1566} - {0.1262}} & {0.0722 - {0.9308}} & {{- 0.1132} + {0.1055}} \\ {0.1311 + {0.6387}} & {{- 0.0029} - {0.3366}} & {{- 0.1758} + {0.1916}} & {{- 0.1479} + {0.6100}} \\ {{- 0.4815} - {0.4045}} & {{- 0.3978} + {0.0753}} & {0.2200 - {0.0040}} & {{- 0.0224} + {0.6259}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.2420 - {0.2055}} & {{- 0.1388} + {0.6904}} & {0.3273 + {0.0199}} \\ {{- 0.3076} - {0.0056}} & {{- 0.5655} - {0.6754}} & {{- 0.2104} + {0.2029}} & {{- 0.1634} + {0.1312}} \\ {0.3448 + {0.0735}} & {0.0981 + {0.0467}} & {0.1060 + {0.1310}} & {{- 0.7933} + {0.4542}} \\ {{- 0.6580} + {0.2285}} & {0.3298 - {0.0517}} & {0.5748 + {0.2447}} & {{- 0.0516} + {0.1018}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.2841} - {0.7969}} & {0.3713 - {0.0745}} & {0.0639 + {0.1460}} \\ {{- 0.0541} + {0.2473}} & {0.0227 + {0.2058`}} & {0.5630 - {0.0230}} & {{- 0.0999} - {0.7520\; }} \\ {0.1311 + {0.6387}} & {0.2505 + {0.3816}} & {0.1527 - {0.2390}} & {0.3715 + {0.3849}} \\ {{- 0.4045} + {0.4815}} & {{- 0.0898} - {0.1576`}} & {{- 0.6437} - {0.2105}} & {0.0144 - {0.3358}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {0.3344 - {0.1640}} & {{- 0.2485} + {0.4791}} & {0.4000 + {0.1590}} \\ {{- 0.5019} + {0.4621}} & {{- 0.1475} + {0.2641}} & {0.5893 + {0.1330}} & {{- 0.1383} - {0.2427`}} \\ {0.2280 + {0.1515}} & {{- 0.1383} + {0.3182}} & {{- 0.4811} + {0.3240}} & {{- 0.5161} - {0.4493}} \\ {{- 0.1515} + {0.2280}} & {{- 0.3105} - {0.7437}} & {{- 0.0811} - {0.0251}} & {{- 0.4817} + {0.1909}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {0.4332 + {0.1612}} & {{- 0.4077} - {0.0486}} & {0.4750 + {0.0865}} \\ {0.5019 - {0.4621}} & {{- 0.2499} + {0.4647}} & {{- 0.3727} - {0.1094}} & {0.2747 + {0.1727}} \\ {0.2280 + {0.1515}} & {0.1294 + {0.5446}} & {0.6827 - {0.1403}} & {0.1161 - {0.3354}} \\ {0.1515 - {0.2280}} & {0.3535 - {0.2640}} & {0.4328 + {0.0863}} & {0.3135 + {0.6613}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {0.5948 - {0.2086}} & {{- 0.6155} + {0.0029}} & {0.0246 + {0.1384}} \\ {0.5606 - {0.2479}} & {{- 0.6101} + {0.1811}} & {{- 0.2226} - {0.2992}} & {0.0747 + {0.2731}} \\ {0.2479 + {0.5606}} & {{- 0.1244} + {0.3625}} & {0.0659 + {0.6219}} & {0.2542 + {0.1473}} \\ {0.2102 - {0.0137}} & {{- 0.1261} + {0.1870}} & {{- 0.1866} + {0.2370}} & {{- 0.7184} - {0.5456}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.7276} + {0.3231}} & {{- 0.0746} + {0.0002}} & {0.1968 + {0.1617}} \\ {0.3076 + {0.0056}} & {0.0623 - {0.3109}} & {0.7114 + {0.3826}} & {0.3902 + {0.0102}} \\ {0.3448 + {0.0735}} & {{- 0.0316} - {0.3607}} & {{- 0.4356} + {0.1774}} & {0.0838 - {0.7186}} \\ {0.6580 - {0.2285}} & {0.1496 - {0.3350}} & {{- 0.1859} - {0.2936}} & {{- 0.1918} + {0.4718}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} & {0.2650 + {0.1831}} & {0.2046 + {0.4515}} & {0.2200 + {0.5535}} \\ {0.3076 + {0.0056}} & {0.0741 + {0.1142}} & {{- 0.2010} + {0.6413}} & {0.4491 - {0.4832}} \\ {0.2285 + {0.6580}} & {0.3746 + {0.4660}} & {0.2087 + {0.1370}} & {{- 0.2423} + {0.1904}} \\ {{- 0.0735} + {0.3448}} & {{- 0.0601} - {0.7188}} & {{- 0.1085} + {0.4782}} & {{- 0.2656} + {0.2110}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {{- 0.3260} - {0.6774}} & {0.4170 + {0.2673}} & {0.0729 - {0.3667}} & {{- 0.1258} + {0.1841}} \\ {0.1709 - {0.3254}} & {{- 0.0605} + {0.0504}} & {{- 0.2440} + {0.1836}} & {0.8489 + {0.2116}} \\ {0.3254 + {0.1709}} & {0.3145 + {0.3209}} & {{- 0.6599} - {0.3142}} & {{- 0.0292} - {0.3579}} \\ {{- 0.4051} - {0.0250}} & {{- 0.4946} + {0.5495}} & {{- 0.3770} + {0.3011}} & {{- 0.2007} + {0.1251}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {{- 0.1499} - {0.0347}} & {{- 0.1568} + {0.5108}} & {{- 0.6900} + {0.1066}} & {0.4419 + {0.0901}} \\ {0.3071 - {0.5009}} & {{- 0.2476} + {0.2774}} & {0.3823 - {0.1521}} & {0.3815 - {0.4492}} \\ {0.1505 + {0.5412}} & {0.1736 - {0.2279}} & {{- 0.1974} - {0.4916}} & {0.3241 - {0.4623}} \\ {{- 0.5412} + {0.1505}} & {{- 0.6443} - {0.2811}} & {0.0584 + {0.2373}} & {0.0318 - {0.3599}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {{- 0.1499} - {0.0347}} & {{- 0.4915} + {0.5944}} & {0.0644 - {0.0145}} & {{- 0.1689} - {0.5903}} \\ {{- 0.3071} + {0.5009}} & {{- 0.1046} - {0.3964}} & {{- 0.5642} + {0.0031}} & {{- 0.3416} - {0.2274}} \\ {0.1505 + {0.5412}} & {0.2288 + {0.3710}} & {{- 0.2561} - {0.0135}} & {0.6471 - {0.0998}} \\ {0.5412 - {0.1505}} & {{- 0.2089} + {0.0583}} & {{- 0.4070} - {0.6677}} & {{- 0.1176} + {0.1097}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {{- 0.0918} - {0.3270}} & {0.0849 - {0.1823}} & {0.5488 + {0.2367}} & {{- 0.3534} - {0.6018}} \\ {{- 0.6387} + {0.1311}} & {{- 0.1210} - {0.0233}} & {{- 0.2347} - {0.3498}} & {0.3248 - {0.5261}} \\ {0.2473 + {0.0541}} & {0.4127 - {0.7969}} & {{- 0.2180} - {0.2794}} & {{- 0.0107} - {0.0700}} \\ {0.4045 - {0.4815}} & {{- 0.0136} + {0.3727}} & {{- 0.3369} - {0.4757}} & {{- 0.2087} - {0.2866}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {{- 0.3841} - {0.3851}} & {0.4828 - {0.2545}} & {0.2119 + {0.3871}} & {{- 0.3824} - {0.2556}} \\ {{- 0.3076} - {0.0056}} & {{- 0.1939} + {0.1069}} & {{- 0.3495} + {0.6691}} & {0.1365 + {0.5175}} \\ {0.2285 + {0.6580}} & {0.3116 + {0.2702}} & {{- 0.1220} + {0.2056}} & {{- 0.5361} - {0.0030}} \\ {0.0735 - {0.3448}} & {{- 0.4258} + {0.5492}} & {0.4208 + {0.0338}} & {{- 0.4460} + {0.1251}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {{- 0.3283} - {0.4339}} & {{- 0.2690} + {0.1335}} & {{- 0.4536} - {0.1528}} \\ {0.2280 + {0.1515}} & {{- 0.3886} - {0.2987}} & {0.0982 - {0.6284}} & {0.5255 + {0.0653}} \\ {0.2280 + {0.1515}} & {0.2457 - {0.2337}} & {{- 0.3618} + {0.5436}} & {0.6122 + {0.0946}} \\ {{- 0.4621} - {0.5019}} & {{- 0.4861} - {0.3354}} & {{- 0.2060} + {0.1911}} & {0.0216 + {0.3261}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.0348} + {0.4484}} & {0.4669 - {0.6204}} & {0.2685 + {0.0858}} \\ {0.4815 + {0.4045}} & {{- 0.4400} + {0.2694}} & {0.0511 + {0.4386}} & {{- 0.1700} + {0.3385}} \\ {0.1311 + {0.6387}} & {0.3429 - {0.0049}} & {0.4475 - {0.0066}} & {0.1480 - {0.4849}} \\ {{- 0.2473} - {0.0541}} & {0.1991 + {0.6118}} & {0.0419 + {0.0073}} & {{- 0.6988} - {0.1782}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.2360} - {0.2973}} & {0.3316 + {0.2303}} & {0.0526 + {0.6279}} \\ {0.2285 + {0.6580}} & {0.0315 - {0.2366}} & {{- 0.3698} - {0.2862}} & {0.3197 - {0.3701}} \\ {0.3448 + {0.0735}} & {0.7285 + {0.0791}} & {0.5225 + {0.0720}} & {{- 0.0255} - {0.2448}} \\ {0.0056 - {0.3076}} & {0.4097 - {0.3067}} & {{- 0.4142} + {0.4106}} & {0.5048 + {0.2198}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {{- 0.5245} - {0.3774}} & {{- 0.5623} - {0.0655}} & {0.2278 + {0.0771}} \\ {{- 0.2102} + {0.0137}} & {{- 0.0522} + {0.4805}} & {{- 0.1132} - {0.5413}} & {0.6451 - {0.0046}} \\ {{- 0.2479} - {0.5606}} & {0.1023 + {0.0593}} & {{- 0.5232} - {0.2730}} & {{- 0.4236} + {0.2873}} \\ {{- 0.5606} + {0.2479}} & {{- 0.5756} - {0.0591}} & {0.0088 - {0.1592}} & {{- 0.3254} - {0.3977}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {0.1953 - {0.3057}} & {{- 0.5062} + {0.0964}} & {0.4500 + {0.1253}} \\ {{- 0.1515} + {0.2280}} & {{- 0.2685} - {0.1895}} & {0.6204 + {0.3044}} & {0.1623 + {0.5596}} \\ {{- 0.2280} - {0.1515}} & {{- 0.7779} - {0.3721}} & {{- 0.3442} - {0.2479}} & {{- 0.0309} + {0.0263}} \\ {{- 0.5019} + {0.4621}} & {{- 0.0216} - {0.1283}} & {0.2771 + {0.0109}} & {{- 0.1897} - {0.6361}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.6035} - {0.3342}} & {{- 0.3380} - {0.3706}} & {{- 0.0686} + {0.3905}} \\ {{- 0.4045} + {0.4815}} & {{- 0.1428} + {0.4838}} & {{- 0.1430} - {0.1006}} & {{- 0.3957} - {0.4036}} \\ {{- 0.1311} - {0.6387}} & {{- 0.1927} + {0.1626}} & {{- 0.0874} + {0.0678}} & {{- 0.6478} + {0.2818}} \\ {{- 0.0541} + {0.2473}} & {{- 0.3638} - {0.2714}} & {0.3209 + {0.7763}} & {{- 0.1557} - {0.0019}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.1371 + {0.7838}} & {0.2479 + {0.0123}} & {{- 0.0008} - {0.0975}} \\ {{- 0.2285} - {0.6580}} & {{- 0.0956} + {0.3657}} & {0.2440 + {0.3447}} & {{- 0.4391} - {0.0271}} \\ {0.3448 + {0.0735}} & {{- 0.1099} - {0.0181}} & {{- 0.2141} + {0.1937}} & {{- 0.2988} + {0.8311}} \\ {{- 0.0056} + {0.3076}} & {0.3423 - {0.3074}} & {0.2247 + {0.7913}} & {{- 0.0349} - {0.1254}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.4422 + {0.0928}} & {{- 0.0860} - {0.2241}} & {0.5338 - {0.0441}} & {{- 0.1033} + {0.6639}} \\ {{- 0.0137} - {0.2102}} & {{- 0.6015} + {0.2527}} & {{- 0.4372} + {0.3957}} & {{- 0.1471} + {0.4008}} \\ {0.2479 + {0.5606}} & {{- 0.4157} - {0.2206}} & {0.0012 + {0.2819}} & {0.4968 - {0.2768}} \\ {0.2479 + {0.5606}} & {0.1543 + {0.5211}} & {{- 0.3515} - {0.4030}} & {{- 0.0551} + {0.1997}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {0.0337 + {0.6193}} & {{- 0.1081} + {0.4460}} & {0.0435 - {0.3239}} & {{- 0.1491} + {0.5251}} \\ {{- 0.2280} - {0.1515}} & {{- 0.3567} - {0.5167}} & {{- 0.1969} - {0.3585}} & {0.3858 + {0.4634}} \\ {0.2280 + {0.1515}} & {{- 0.2354} - {0.1914}} & {0.2532 + {0.7745}} & {0.0863 + {0.4020}} \\ {0.4621 + {0.5019}} & {{- 0.1985} - {0.5136}} & {0.0895 - {0.2322}} & {{- 0.0884} - {0.4019}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {0.0918 + {0.3270}} & {{- 0.5060} + {0.1443}} & {0.5534 - {0.3836}} & {{- 0.1155} - {0.3757}} \\ {0.4045 - {0.4815}} & {{- 0.4682} - {0.5571}} & {{- 0.1369} - {0.0572}} & {{- 0.2290} + {0.0234}} \\ {{- 0.1311} - {0.6387}} & {0.4345 + {0.0089}} & {0.5727 - {0.1978}} & {{- 0.1012} - {0.0932}} \\ {0.0541 - {0.2473}} & {{- 0.0309} + {0.0618}} & {{- 0.2903} - {0.2707}} & {0.7177 - {0.5085}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.0427 + {0.4037}} & {0.1038 + {0.3577}} & {0.5172 + {0.3648}} \\ {0.6580 - {0.2285}} & {{- 0.3221} + {0.3130}} & {{- 0.3384} - {0.3674}} & {{- 0.2477} + {0.0474}} \\ {{- 0.3448} - {0.0735}} & {{- 0.0724} + {0.1313}} & {{- 0.0231} - {0.7092}} & {0.5412 + {0.2385}} \\ {0.3076 + {0.0056}} & {0.3259 - {0.7104}} & {{- 0.3205} - {0.0747}} & {0.1184 + {0.4148}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.4422 + {0.1928}} & {{- 0.1512} + {0.0220}} & {{- 0.5578} + {0.0360}} & {{- 0.4803} + {0.4790}} \\ {0.2102 - {0.0137}} & {{- 0.5297} - {0.2125}} & {{- 0.4245} + {0.2405}} & {0.3258 - {0.5345}} \\ {{- 0.2479} - {0.5606}} & {0.3488 + {0.2574}} & {{- 0.5660} - {0.2718}} & {{- 0.0021} - {0.2051}} \\ {0.5606 - {0.2479}} & {0.5345 - {0.4211}} & {0.1552 + {0.1767}} & {{- 0.1657} - {0.2802}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.3899 + {0.5099}} & {0.1239 + {0.3480}} & {{- 0.3212} + {0.2294}} \\ {0.0022 + {0.1690}} & {{- 0.2642} - {0.2730}} & {{- 0.0709} + {0.7215}} & {0.4401 + {0.3283}} \\ {{- 0.3076} - {0.0056}} & {0.0634 + {0.5454}} & {{- 0.0400} - {0.3119}} & {0.5426 + {0.4590}} \\ {{- 0.7536} - {0.1140}} & {0.3743 - {0.0464}} & {0.3735 + {0.3156}} & {{- 0.1921} - {0.0291}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} & {0.1996 + {0.3349}} & {0.3567 - {0.6608}} & {0.1085 + {0.0728}} \\ {0.0837 + {0.4175}} & {0.1647 + {0.4478}} & {{- 0.0991} + {0.3844}} & {{- 0.5490} - {0.3633}} \\ {{- 0.4597} + {0.3989}} & {0.0029 - {0.0825}} & {{- 0.4340} + {0.1941}} & {{- 0.0351} + {0.6288}} \\ {{- 0.4175} + {0.0837}} & {0.6245 - {0.4727}} & {0.1497 + {0.1732}} & {0.1381 - {0.3657}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.4945 - {0.2660}} & {{- 0.4949} + {0.3789}} & {{- 0.0039} - {0.0184}} \\ {{- 0.1140} + {0.7536}} & {{- 0.4531} - {0.0810}} & {0.2332 + {0.1714}} & {{- 0.0493} - {0.3478}} \\ {{- 0.3076} - {0.0056}} & {0.2295 + {0.4493}} & {0.2178 + {0.6234}} & {{- 0.4191} + {0.1977}} \\ {{- 0.1690} + {0.0022}} & {0.0234 - {0.4666}} & {{- 0.0755} - {0.2931}} & {{- 0.8112} + {0.0581}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {0.3396 + {0.1614}} & {{- 0.0891} - {0.1737`}} & {{- 0.5874} + {0.0189}} & {0.4181 + {0.5480}} \\ {{- 0.3400} - {0.3060}} & {{- 0.2160} + {0.0403}} & {{- 0.4608} - {0.5955}} & {0.2155 - {0.3594}} \\ {0.3493 - {0.5641}} & {0.3530 + {0.5446}} & {{- 0.2058} - {0.0338}} & {{- 0.2629} + {0.1610}} \\ {{- 0.3060} + {0.3400}} & {0.1657 + {0.6818}} & {0.1876 - {0.0948}} & {0.4723 + {0.1766}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.3162 - {0.1330}} & {{- 0.5287} - {0.1679}} & {0.1564 + {0.5043}} \\ {{- 0.1690} + {0.0022}} & {{- 0.2451} + {0.4651}} & {0.2831 + {0.1482}} & {0.4827 + {0.6000}} \\ {0.3076 + {0.0056}} & {0.1535 - {0.5047}} & {0.7325 - {0.1862}} & {0.0058 + {0.2362}} \\ {{- 0.1140} + {0.7536}} & {{- 0.4972} - {0.2836}} & {0.0497 + {0.1282}} & {0.1692 - {0.2094}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} & {0.1157 + {0.3948}} & {0.5787 - {0.3292}} & {0.2943 - {0.1843}} \\ {{- 0.4175} + {0.0837}} & {0.2814 - {0.3175}} & {0.2525 - {0.2620}} & {{- 0.5641} + {0.4338}} \\ {0.4597 - {0.3989}} & {0.4393 + {0.0781}} & {0.3512 + {0.0363}} & {0.2819 + {0.4758}} \\ {0.0837 + {0.4175}} & {0.3704 - {0.5609}} & {0.1336 + {0.5309}} & {0.1521 - {0.2100}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.6183} - {0.3893}} & {{- 0.3104} - {0.0769}} & {0.2262 - {0.1301}} \\ {0.1140 - {0.7536}} & {0.0439 - {0.4301}} & {0.0424 - {0.0174}} & {0.4717 - {0.0872`}} \\ {{- 0.3076} - {0.0056}} & {{- 0.3231} + {0.0445}} & {{- 0.0056} + {0.4980}} & {0.2818 + {0.6867}} \\ {0.1690 - {0.0022}} & {{- 0.2461} - {0.3351}} & {0.7828 + {0.1865}} & {{- 0.3883} + {0.0121}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.3396 + {0.1614}} & {0.3566 - {0.2418}} & {0.0169 - {0.4765}} & {{- 0.5447} - {0.3859}} \\ {0.3060 - {0.3400}} & {{- 0.1924} + {0.0246}} & {{- 0.3480} + {0.4512}} & {0.0113 - {0.6545}} \\ {{- 0.3493} + {0.5641}} & {{- 0.5527} + {0.1626}} & {{- 0.3128} - {0.0721}} & {{- 0.3061} - {0.1765}} \\ {0.3400 + {0.3060}} & {{- 0.4030} - {0.5315}} & {0.5097 + {0.2917}} & {0.0150 + {0.0288}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {0.0072 + {0.6080}} & {0.0164 + {0.4489}} & {0.2019 + {0.3032}} \\ {{- 0.0022} - {0.1690}} & {{- 0.1531} - {0.3405}} & {{- 0.1388} + {0.8192}} & {0.0946 - {0.3643}} \\ {{- 0.3076} - {0.0056}} & {0.4820 + {0.4643}} & {{- 0.2314} + {0.1231}} & {{- 0.4842} + {0.3928}} \\ {0.7536 + {0.1140}} & {{- 0.1738} - {0.1132}} & {{- 0.1247} - {0.1538}} & {{- 0.4937} - {0.3051}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {{- 0.1560} + {0.4926}} & {{- 0.4387} - {0.0343}} & {0.5100 - {0.0764}} & {0.2952 + {0.4317}} \\ {0.4175 - {0.0837`}} & {0.5062 - {0.5972}} & {0.2808 - {0.2160}} & {0.1570 + {0.2361}} \\ {0.4597 - {0.3989}} & {{- 0.2611} + {0.2853}} & {{- 0.1195} + {0.0730}} & {{- 0.1784} + {0.6546}} \\ {{- 0.0837} - {0.4175}} & {{- 0.0591} + {0.2011}} & {0.7066 - {0.2995}} & {{- 0.3568} - {0.2417}} \end{matrix}$

${{V4}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.3841 + {0.3851}} & {{- 0.0757} - {0.5813}} & {0.1457 - {0.0933}} & {{- 0.5106} + {0.2643}} \\ {0.7536 + {0.1140}} & {0.1019 + {0.2454}} & {0.3241 + {0.2186}} & {0.2092 - {0.3897}} \\ {0.3076 + {0.0056}} & {0.5989 + {0.2606}} & {{- 0.5728} - {0.0137}} & {{- 0.1198} + {0.3689}} \\ {{- 0.0022} - {0.1690}} & {{- 0.3284} + {0.2264}} & {{- 0.1976} + {0.6708}} & {{- 0.5620} - {0.0868}} \end{matrix}$ ${{V4}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.3396 + {0.1614}} & {0.2711 - {0.7070}} & {{- 0.2911} + {0.2527}} & {{- 0.1768} + {0.3248}} \\ {0.3400 + {0.3060}} & {{- 0.4123} - {0.1234}} & {0.0110 - {0.0641}} & {{- 0.4612} - {0.6234}} \\ {0.3493 - {0.5641}} & {0.4714 - {0.0224}} & {0.3098 + {0.2417}} & {0.0189 - {0.4271}} \\ {0.3060 - {0.3400}} & {{- 0.0617} - {0.1224}} & {{- 0.4468} - {0.7023}} & {0.2720 - {0.0718}} \end{matrix}$

5. Fifth Scheme to Design a 6-Bit Codebook

According to an exemplary embodiment, a 4-bit codebook and a 6-bit codebook may be provided in an integrated form.

(1) Operation 1:

16 matrices may be given as follows:

${W\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$ ${W\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${W\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${W\left( {\text{:},\text{:},4} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000`} & {- 0.5000} \\ {{- 0.4985} - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

${W\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.1913 + {0.4619}} & {0.4619 - {0.1913}} & {{- 0.1913} - {0.4619}} & {{- 0.4619} + {0.1913}} \end{matrix}$ ${W\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619}} & {{- 0.4619} + {0.1913}} & {{- 0.1913} - {0.4619}} & {0.4619 - {0.1913}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.4619} - {0.1913}} & {{- 0.1913} + {0.4619}} & {0.4619 + {0.1913}} & {0.1913 - {0.4619}} \end{matrix}$ ${W\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.4258 + {0.0076}} & {0.5210 - {0.4265}} & {{- 0.2364} - {0.1268}} \\ {0.5740 - {0.3102}} & {0.0596 + {0.3222}} & {0.2922 + {0.0454}} & {0.6147 + {0.0409}} \\ {0.2823 - {0.3783}} & {{- 0.4186} + {0.1136}} & {0.3628 + {0.1522}} & {{- 0.6500} - {0.1085}} \\ {0.2062 + {0.1248}} & {0.1513 + {0.7073}} & {{- 0.5498} - {0.0464}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${W\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.0742 + {0.2714}} & {0.3370 + {0.2000}} & {{- 0.1041} - {0.7125}} \\ {0.5545 + {0.3908}} & {{- 0.0591} - {0.0963}} & {0.2158 + {0.4089}} & {{- 0.5589} + {0.0299}} \\ {0.2593 - {0.0234}} & {{- 0.5800} - {0.1645}} & {{- 0.2363} - {0.5965}} & {{- 0.2149} - {0.3331}} \\ {0.3300 + {0.3380}} & {0.4619 + {0.5755}} & {0.0175 - {0.4698}} & {0.0748 - {0.0748}} \end{matrix}$

${W\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.2121 - {0.4391}} & {0.4649 + {0.1545}} & {{- 0.4709} + {0.0383}} \\ {0.0570 + {0.3132}} & {0.3637 + {0.1907}} & {0.5971 - {0.5122}} & {0.3285 + {0.0569}} \\ {0.2569 - {0.1797}} & {0.0539 - {0.3157}} & {{- 0.0025} + {0.3671}} & {0.8042 + {0.1327}} \\ {0.6949 + {0.1358}} & {{- 0.5835} + {0.3879}} & {0.0058 - {0.0797}} & {0.0341 + {0.0124}} \end{matrix}$ ${W\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.2199 + {0.3199}} & {0.6305 - {0.4345}} & {{- 0.0525} + {0.2357}} & {{- 0.3110} - {0.3286}} \\ {0.5983 + {0.1264}} & {{- 0.0683} + {0.2985}} & {0.5833 + {0.2614}} & {0.2969 - {0.1888}} \\ {{- 0.5576} - {0.2903}} & {{- 0.0495} - {0.3235}} & {0.3552 + {0.3750}} & {0.2888 - {0.3841}} \\ {{- 0.0940} - {0.2672}} & {{- 0.0481} + {0.4588}} & {0.0118 + {0.5160}} & {{- 0.6633} - {0.0260}} \end{matrix}$ ${W\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.2632 + {0.4348}} & {0.1838 + {0.6429}} & {0.2307 + {0.1558}} \\ {{- 0.0163} + {0.7556}} & {0.3698 - {0.0222}} & {0.0055 - {0.3112}} & {0.3467 - {0.2728}} \\ {0.0681 + {0.3202}} & {{- 0.4066} - {0.6110}} & {0.3882 + {0.2198}} & {0.1705 + {0.3551}} \\ {0.1259 + {0.2977}} & {0.0347 + {0.2542}} & {0.4482 - {0.2369}} & {{- 0.7319} + {0.1924}} \end{matrix}$ ${W\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.4493} + {0.3172}} & {{- 0.3346} + {0.1479}} & {{- 0.4868} - {0.0811}} \\ {{- 0.1472} - {0.1184}} & {{- 0.5422} + {0.3203}} & {0.2302 - {0.4770}} & {0.4913 + {0.2140}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1168} - {0.4073}} & {{- 0.1288} - {0.3642}} & {{- 0.2957} - {0.1634}} \\ {0.2881 - {0.0638}} & {0.3438 + {0.0562}} & {{- 0.1358} - {0.6465}} & {{- 0.3579} + {0.4766}} \end{matrix}$

${W\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.6462 - {0.3049}} & {0.1765 - {0.0924}} & {0.3680 + {0.1691}} \\ {{- 0.2340} + {0.0529}} & {{- 0.0388} - {0.4789}} & {0.6212 - {0.0425}} & {{- 0.5223} + {0.2262}} \\ {0.0491 - {0.2199}} & {{- 0.2962} + {0.1142}} & {0.5130 + {0.5000}} & {0.5366 + {0.2175}} \\ {{- 0.7746} + {0.0772}} & {0.1089 - {0.3821}} & {{- 0.2395} + {0.0459}} & {0.4196 + {0.0255}} \end{matrix}$ ${W\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0848 + {0.2248`}} & {0.5883 + {0.1906}} & {{- 0.4550} - {0.1147}} & {{- 0.5202} + {0.2628}} \\ {{- 0.2010} + {0.7357}} & {{- 0.0481} - {0.2343}} & {{- 0.3923} + {0.0511}} & {0.3107 - {0.3288}} \\ {0.3862 + {0.2270}} & {0.2323 + {0.5375}} & {0.1654 - {0.1078}} & {0.6089 + {0.2161}} \\ {{- 0.3114} + {0.2509}} & {0.4661 + {0.0138}} & {0.6358 + {0.4245}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${W\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {0.3968 - {0.0671}} & {0.3110 + {0.1356}} & {{- 0.2168} - {0.4540}} & {{- 0.6008} - {0.3299}} \\ {{- 0.8289} - {0.0667}} & {{- 0.2015} + {0.0416}} & {0.1102 - {0.2871}} & {{- 0.4011} - {0.1031}} \\ {0.1770 - {0.1687}} & {{- 0.3325} - {0.1370}} & {0.0567 - {0.7634}} & {0.2785 + {0.3837}} \\ {{- 0.2814} + {0.0866}} & {0.7347 - {0.4165}} & {{- 0.0390} - {0.2544}} & {0.3519 - {0.1000}} \end{matrix}$ ${W\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {0.0743 - {0.7846}} & {{- 0.4063} + {0.3394}} & {{- 0.2621} + {0.0798}} & {0.1421 + {0.0583}} \\ {0.1307 - {0.3436}} & {0.2466 + {0.0922}} & {0.4783 - {0.5634}} & {{- 0.0799} - {0.4928}} \\ {{- 0.3387} - {0.2385}} & {{- 0.4080} - {0.5055}} & {0.5212 + {0.1951}} & {{- 0.2917} + {0.1078}} \\ {{- 0.0962} + {0.2508}} & {{- 0.4767} - {0.0347}} & {{- 0.2418} + {0.1026}} & {{- 0.0231} - {0.7937}} \end{matrix}$

(2) Operation 2:

According to an exemplary embodiment, a 4-bit rank 1 codebook, a rank 2 codebook, a rank 3 codebook, and a rank 4 codebook, and a 6-bit rank 1 codebook, a rank 2 codebook, a rank 3 codebook, and a rank 4 codebook may be generated as given by the following Table 15 and Table 16, using column subsets of the aforementioned 16 matrices. The following Table 15 shows 6 bits of codebooks:

Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 W(:,1,1) W(:,[1,2],1) W(:,[1,2,3],1) W(:,:,1) 2 W(:,2,1) W(:,[1,3],1) W(:,[1,2,4],1) W(:,:,2) 3 W(:,3,1) W(:,[1,4],1) W(:,[1,3,4],1) W(:,:,3) 4 W(:,4,1) W(:,[2,3],1) W(:,[2,3,4],1) W(:,:,4) 5 W(:,1,2) W(:,[2,4],1) W(:,[1,2,3],2) W(:,:,5) 6 W(:,2,2) W(:,[3,4],1) W(:,[1,2,4],2) W(:,:,6) 7 W(:,3,2) W(:,[1,2],2) W(:,[1,3,4],2) W(:,:,7) 8 W(:,4,2) W(:,[1,3],2) W(:,[2,3,4],2) W(:,:,8) 9 W(:,1,3) W(:,[1,4],2) W(:,[1,2,3],3) W(:,:,9) 10 W(:,2,3) W(:,[2,3],2) W(:,[1,2,4],3) W(:,:,10) 11 W(:,3,3) W(:,[2,4],2) W(:,[1,3,4],3) W(:,:,11) 12 W(:,4,3) W(:,[3,4],2) W(:,[2,3,4],3) W(:,:,12) 13 W(:,1,4) W(:,[1,4],3) W(:,[1,2,3],4) W(:,:,13) 14 W(:,2,4) W(:,[2,3],3) W(:,[1,2,4],4) W(:,:,14) 15 W(:,3,4) W(:,[1,2],4) W(:,[1,3,4],4) W(:,:,15) 16 W(:,4,4) W(:,[1,4],4) W(:,[2,3,4],4) W(:,:,16) 17 W(:,1,5) W(:,[1,2],5) W(:,[1,2,3],5) 18 W(:,2,5) W(:,[1,3],6) W(:,[1,2,4],5) 19 W(:,3,5) W(:,[2,3],6) W(:,[1,3,4],5) 20 W(:,4,5) W(:,[2,4],6) W(:,[2,3,4],5) 21 W(:,1,6) W(:,[1,2],7) W(:,[1,2,3],6) 22 W(:,2,6) W(:,[1,3],7) W(:,[1,2,4],6) 23 W(:,3,6) W(:,[1,4],7) W(:,[1,3,4],6) 24 W(:,4,6) W(:,[2,3],7) W(:,[2,3,4],6) 25 W(:,1,7) W(:,[2,4],7) W(:,[1,2,3],7) 26 W(:,2,7) W(:,[3,4],7) W(:,[1,2,4],7) 27 W(:,3,7) W(:,[1,2],8) W(:,[1,3,4],7) 28 W(:,4,7) W(:,[1,3],8) W(:,[2,3,4],7) 29 W(:,1,8) W(:,[1,4],8) W(:,[1,2,3],8) 30 W(:,2,8) W(:,[2,3],8) W(:,[1,2,4],8) 31 W(:,3,8) W(:,[2,4],8) W(:,[1,3,4],8) 32 W(:,4,8) W(:,[3,4],8) W(:,[2,3,4],8) 32 W(:,1,9) W(:,[1,2],9) W(:,[1,2,3],9) 34 W(:,2,9) W(:,[1,3],9) W(:,[1,2,4],9) 35 W(:,3,9) W(:,[2,3],9) W(:,[1,3,4],9) 36 W(:,4,9) W(:,[2,4],9) W(:,[2,3,4],9) 37 W(:,1,10) W(:,[1,3],10) W(:,[1,2,3],10) 38 W(:,2,10) W(:,[2,4],10) W(:,[1,2,4],10) 39 W(:,3,10) W(:,[3,4],10) W(:,[1,3,4],10) 40 W(:,4,10) W(:,[1,2],11) W(:,[2,3,4],10) 41 W(:,1,11) W(:,[1,3],11) W(:,[1,2,3],11) 42 W(:,2,11) W(:,[1,4],11) W(:,[1,2,4],11) 43 W(:,3,11) W(:,[2,3],11) W(:,[1,3,4],11) 44 W(:,4,11) W(:,[2,4],11) W(:,[2,3,4],11) 45 W(:,1,12) W(:,[3,4],11) W(:,[1,2,3],12) 46 W(:,2,12) W(:,[1,2],12) W(:,[1,2,4],12) 47 W(:,3,12) W(:,[1,3],12) W(:,[1,3,4],12) 48 W(:,4,12) W(:,[2,3],12) W(:,[2,3,4],12) 49 W(:,1,13) W(:,[1,2],13) W(:,[1,2,3],13) 50 W(:,2,13) W(:,[1,4],13) W(:,[1,2,4],13) 51 W(:,3,13) W(:,[2,4],13) W(:,[1,3,4],13) 52 W(:,4,13) W(:,[3,4],13) W(:,[2,3,4],13) 53 W(:,1,14) W(:,[1,2],14) W(:,[1,2,3],14) 54 W(:,2,14) W(:,[1,3],14) W(:,[1,2,4],14) 55 W(:,3,14) W(:,[1,4],14) W(:,[1,3,4],14) 56 W(:,4,14) W(:,[2,3],14) W(:,[2,3,4],14) 57 W(:,1,15) W(:,[2,4],14) W(:,[1,2,3],15) 58 W(:,2,15) W(:,[3,4],14) W(:,[1,2,4],15) 59 W(:,3,15) W(:,[1,4],15) W(:,[1,3,4],15) 60 W(:,4,15) W(:,[2,3],15) W(:,[2,3,4],15) 61 W(:,1,16) W(:,[1,3],16) W(:,[1,2,3],16) 62 W(:,2,16) W(:,[2,3],16) W(:,[1,2,4],16) 63 W(:,3,16) W(:,[2,4],16) W(:,[1,3,4],16) 64 W(:,4,16) W(:,[3,4],16) W(:,[2,3,4],16)

The following Table 16 shows 4 bits of codebooks:

Codebook Transmission Transmission Transmission Transmission Index Rank 1 Rank 2 Rank 3 Rank 4 1 W(:,1,1) W(:,[1,2],1) W(:,[1,2,3],1) W(:,:,1) 2 W(:,2,1) W(:,[1,3],1) W(:,[1,2,4],1) W(:,:,2) 3 W(:,3,1) W(:,[1,4],1) W(:,[1,3,4],1) W(:,:,3) 4 W(:,4,1) W(:,[2,3],1) W(:,[2,3,4],1) W(:,:,4) 5 W(:,1,2) W(:,[2,4],1) W(:,[1,2,3],2) 6 W(:,2,2) W(:,[3,4],1) W(:,[1,2,4],2) 7 W(:,3,2) W(:,[1,2],2) W(:,[1,3,4],2) 8 W(:,4,2) W(:,[1,3],2) W(:,[2,3,4],2) 9 W(:,1,3) W(:,[1,4],2) W(:,[1,2,3],3) 10 W(:,2,3) W(:,[2,3],2) W(:,[1,2,4],3) 11 W(:,3,3) W(:,[2,4],2) W(:,[1,3,4],3) 12 W(:,4,3) W(:,[3,4],2) W(:,[2,3,4],3) 13 W(:,1,4) W(:,[1,4],3) W(:,[1,2,3],4) 14 W(:,2,4) W(:,[2,3],3) W(:,[1,2,4],4) 15 W(:,3,4) W(:,[1,2],4) W(:,[1,3,4],4) 16 W(:,4,4) W(:,[1,4],4) W(:,[2,3,4],4)

Codes according to a MATLAB® program for obtaining the 6-bit codebooks disclosed in the above Table 15 may follow as:

6-Bit Rank 1 Codebook:

Codebook{1}(:,1,1:4)=W(:,[1:4],1); Codebook{1}(:,1,5:8)=W(:,[1:4],2); Codebook{1}(:,1,9:12)=W(:,[1:4],3); Codebook{1}(:,1,13:16)=W(:,[1:4],4); Codebook{1}(:,1,17:20)=W(:,[1:4],5); Codebook{1}(:,1,21:24)=W(:,[1:4],6); Codebook{1}(:,1,25:28)=W(:,[1:4],7); Codebook{1}(:,1,29:32)=W(:,[1:4],8); Codebook{1}(:,1,33:36)=W(:,[1:4],9); Codebook{1}(:,1,37:40)=W(:,[1:4],10); Codebook{1}(:,1,41:44)=W(:,[1:4],11); Codebook{1}(:,1,45:48)=W(:,[1:4],12); Codebook{1}(:,1,49:52)=W(:,[1:4],13); Codebook{1}(:,1,53:56)=W(:,[1:4],14); Codebook{1}(:,1,57:60)=W(:,[1:4],15); Codebook{1}(:,1,61:64)=W(:,[1:4],16);

6-Bit Rank 2 Codebook:

Codebook{2}(:,1:2,1)=W(:,[1,2],1); Codebook{2}(:,1:2,2)=W(:,[1,3],1); Codebook{2}(:,1:2,3)=W(:,[1,4],1); Codebook{2}(:,1:2,4)=W(:,[2,3],1); Codebook{2}(:,1:2,5)=W(:,[2,4],1); Codebook{2}(:,1:2,6)=W(:,[3,4],1); Codebook{2}(:,1:2,7)=W(:,[1,2],2); Codebook{2}(:,1:2,8)=W(:,[1,3],2); Codebook{2}(:,1:2,9)=W(:,[1,4],2); Codebook{2}(:,1:2,10)=W(:,[2,3],2); Codebook{2}(:,1:2,11)=W(:,[2,4],2); Codebook{2}(:,1:2,12)=W(:,[3,4],2); Codebook{2}(:,1:2,13)=W(:,[1,4],3); Codebook{2}(:,1:2,14)=W(:,[2,3],3); Codebook{2}(:,1:2,15)=W(:,[1,2],4); Codebook{2}(:,1:2,16)=W(:,[1,4],4); Codebook{2}(:,1:2,17)=W(:,[1,2],5); Codebook{2}(:,1:2,18)=W(:,[1,3],6); Codebook{2}(:,1:2,19)=W(:,[2,3],6); Codebook{2}(:,1:2,20)=W(:,[2,4],6); Codebook{2}(:,1:2,21)=W(:,[1,2],7); Codebook{2}(:,1:2,22)=W(:,[1,3],7); Codebook{2}(:,1:2,23)=W(:,[1,4],7); Codebook{2}(:,1:2,24)=W(:,[2,3],7); Codebook{2}(:,1:2,25)=W(:,[2,4],7); Codebook{2}(:,1:2,26)=W(:,[3,4],7); Codebook{2}(:,1:2,27)=W(:,[1,2],8); Codebook{2}(:,1:2,28)=W(:,[1,3],8); Codebook{2}(:,1:2,29)=W(:,[1,4],8); Codebook{2}(:,1:2,30)=W(:,[2,3],8); Codebook{2}(:,1:2,31)=W(:,[2,4],8); Codebook{2}(:,1:2,32)=W(:,[3,4],8); Codebook{2}(:,1:2,33)=W(:,[1,2],9); Codebook{2}(:,1:2,34)=W(:,[1,3],9); Codebook{2}(:,1:2,35)=W(:,[2,3],9); Codebook{2}(:,1:2,36)=W(:,[2,4],9); Codebook{2}(:,1:2,37)=W(:,[1,3],10); Codebook{2}(:,1:2,38)=W(:,[2,4],10); Codebook{2}(:,1:2,39)=W(:,[3,4],10); Codebook{2}(:,1:2,40)=W(:,[1,2],11); Codebook{2}(:,1:2,41)=W(:,[1,3],11); Codebook{2}(:,1:2,42)=W(:,[1,4],11); Codebook{2}(:,1:2,43)=W(:,[2,3],11); Codebook{2}(:,1:2,44)=W(:,[2,4],11); Codebook{2}(:,1:2,45)=W(:,[3,4],11); Codebook{2}(:,1:2,46)=W(:,[1,2],12); Codebook{2}(:,1:2,47)=W(:,[1,3],12); Codebook{2}(:,1:2,48)=W(:,[2,3],12); Codebook{2}(:,1:2,49)=W(:,[1,2],13); Codebook{2}(:,1:2,50)=W(:,[1,4],13); Codebook{2}(:,1:2,51)=W(:,[2,4],13); Codebook{2}(:,1:2,52)=W(:,[3,4],13); Codebook{2}(:,1:2,53)=W(:,[1,2],14); Codebook{2}(:,1:2,54)=W(:,[1,3],14); Codebook{2}(:,1:2,55)=W(:,[1,4],14); Codebook{2}(:,1:2,56)=W(:,[2,3],14); Codebook{2}(:,1:2,57)=W(:,[2,4],14); Codebook{2}(:,1:2,58)=W(:,[3,4],14); Codebook{2}(:,1:2,59)=W(:,[1,4],15); Codebook{2}(:,1:2,60)=W(:,[2,3],15); Codebook{2}(:,1:2,61)=W(:,[1,3],16); Codebook{2}(:,1:2,62)=W(:,[2,3],16); Codebook{2}(:,1:2,63)=W(:,[2,4],16); Codebook{2}(:,1:2,64)=W(:,[3,4],16);

6-Bit Rank 3 Codebook:

Codebook{3}(:,1:3,1)=W(:,[1,2,3],1); Codebook{3}(:,1:3,2)=W(:,[1,2,4],1); Codebook{3}(:,1:3,3)=W(:,[1,3,4],1); Codebook{3}(:,1:3,4)=W(:,[2,3,4],1); Codebook{3}(:,1:3,5)=W(:,[1,2,3],2); Codebook{3}(:,1:3,6)=W(:,[1,2,4],2); Codebook{3}(:,1:3,7)=W(:,[1,3,4],2); Codebook{3}(:,1:3,8)=W(:,[2,3,4],2); Codebook{3}(:,1:3,9)=W(:,[1,2,3],3); Codebook{3}(:,1:3,10)=W(:,[1,2,4],3); Codebook{3}(:,1:3,11)=W(:,[1,3,4],3); Codebook{3}(:,1:3,12)=W(:,[2,3,4],3); Codebook{3}(:,1:3,13)=W(:,[1,2,3],4); Codebook{3}(:,1:3,14)=W(:,[1,2,4],4); Codebook{3}(:,1:3,15)=W(:,[1,3,4],4); Codebook{3}(:,1:3,16)=W(:,[2,3,4],4); Codebook{3}(:,1:3,17)=W(:,[1,2,3],5); Codebook{3}(:,1:3,18)=W(:,[1,2,4],5); Codebook{3}(:,1:3,19)=W(:,[1,3,4],5); Codebook{3}(:,1:3,20)=W(:,[2,3,4],5); Codebook{3}(:,1:3,21)=W(:,[1,2,3],6); Codebook{3}(:,1:3,22)=W(:,[1,2,4],6); Codebook{3}(:,1:3,23)=W(:,[1,3,4],6); Codebook{3}(:,1:3,24)=W(:,[2,3,4],6); Codebook{3}(:,1:3,25)=W(:,[1,2,3],7); Codebook{3}(:,1:3,26)=W(:,[1,2,4],7); Codebook{3}(:,1:3,27)=W(:,[1,3,4],7); Codebook{3}(:,1:3,28)=W(:,[2,3,4],7); Codebook{3}(:,1:3,29)=W(:,[1,2,3],8); Codebook{3}(:,1:3,30)=W(:,[1,2,4],8); Codebook{3}(:,1:3,31)=W(:,[1,3,4],8); Codebook{3}(:,1:3,32)=W(:,[2,3,4],8); Codebook{3}(:,1:3,33)=W(:,[1,2,3],9); Codebook{3}(:,1:3,34)=W(:,[1,2,4],9); Codebook{3}(:,1:3,35)=W(:,[1,3,4],9); Codebook{3}(:,1:3,36)=W(:,[2,3,4],9); Codebook{3}(:,1:3,37)=W(:,[1,2,3],10); Codebook{3}(:,1:3,38)=W(:,[1,2,4],10); Codebook{3}(:,1:3,39)=W(:,[1,3,4],10); Codebook{3}(:,1:3,40)=W(:,[2,3,4],10); Codebook{3}(:,1:3,41)=W(:,[1,2,3],11); Codebook{3}(:,1:3,42)=W(:,[1,2,4],11); Codebook{3}(:,1:3,43)=W(:,[1,3,4],11); Codebook{3}(:,1:3,44)=W(:,[2,3,4],11); Codebook{3}(:,1:3,45)=W(:,[1,2,3],12); Codebook{3}(:,1:3,46)=W(:,[1,2,4],12); Codebook{3}(:,1:3,47)=W(:,[1,3,4],12); Codebook{3}(:,1:3,48)=W(:,[2,3,4],12); Codebook{3}(:,1:3,49)=W(:,[1,2,3],13); Codebook{3}(:,1:3,50)=W(:,[1,2,4],13); Codebook{3}(:,1:3,51)=W(:,[1,3,4],13); Codebook{3}(:,1:3,52)=W(:,[2,3,4],13); Codebook{3}(:,1:3,53)=W(:,[1,2,3],14); Codebook{3}(:,1:3,54)=W(:,[1,2,4],14); Codebook{3}(:,1:3,55)=W(:,[1,3,4],14); Codebook{3}(:,1:3,56)=W(:,[2,3,4],14); Codebook{3}(:,1:3,57)=W(:,[1,2,3],15); Codebook{3}(:,1:3,58)=W(:,[1,2,4],15); Codebook{3}(:,1:3,59)=W(:,[1,3,4],15); Codebook{3}(:,1:3,60)=W(:,[2,3,4],15); Codebook{3}(:,1:3,61)=W(:,[1,2,3],16); Codebook{3}(:,1:3,62)=W(:,[1,2,4],16); Codebook{3}(:,1:3,63)=W(:,[1,3,4],16); Codebook{3}(:,1:3,64)=W(:,[2,3,4],16);

6-Bit Rank 4 Codebook:

Codebook{4}(:,1:4,1)=W(:,[1,2,3,4],1); Codebook{4}(:,1:4,2)=W(:,[1,2,3,4],2); Codebook{4}(:,1:4,3)=W(:,[1,2,3,4],3); Codebook{4}(:,1:4,4)=W(:,[1,2,3,4],4); Codebook{4}(:,1:4,5)=W(:,[1,2,3,4],5); Codebook{4}(:,1:4,6)=W(:,[1,2,3,4],6); Codebook{4}(:,1:4,7)=W(:,[1,2,3,4],7); Codebook{4}(:,1:4,8)=W(:,[1,2,3,4],8); Codebook{4}(:,1:4,9)=W(:,[1,2,3,4],9); Codebook{4}(:,1:4,10)=W(:,[1,2,3,4],10); Codebook{4}(:,1:4,11)=W(:,[1,2,3,4],11); Codebook{4}(:,1:4,12)=W(:,[1,2,3,4],12); Codebook{4}(:,1:4,13)=W(:,[1,2,3,4],13); Codebook{4}(:,1:4,14)=W(:,[1,2,3,4],14); Codebook{4}(:,1:4,15)=W(:,[1,2,3,4],15); Codebook{4}(:,1:4,16)=W(:,[1,2,3,4],16);

Codes according to the MATLAB® program for obtaining the 4-bit codebooks disclosed in the above Table 16 may follow as:

4-Bit Rank 1 Codebook:

Codebook{1}(:,1,1:4)=W(:,[l:4],1); Codebook{1}(:,1,5:8)=W(:,[l:4],2); Codebook{1}(:,1,9:12)=W(:,[l:4],3); Codebook{1}(:,1,13:16)=W(:,[l:4],4);

4-Bit Rank 2 Codebook:

Codebook{2}(:,1:2,1)=W(:,[1,2],1); Codebook{2}(:,1:2,2)=W(:,[1,3],1); Codebook{2}(:,1:2,3)=W(:,[1,4],1); Codebook{2}(:,1:2,4)=W(:,[2,3],1); Codebook{2}(:,1:2,5)=W(:,[2,4],1); Codebook{2}(:,1:2,6)=W(:,[3,4],1); Codebook{2}(:,1:2,7)=W(:,[1,2],2); Codebook{2}(:,1:2,8)=W(:,[1,3],2); Codebook{2}(:,1:2,9)=W(:,[1,4],2); Codebook{2}(:,1:2,10)=W(:,[2,3],2); Codebook{2}(:,1:2,11)=W(:,[2,4],2); Codebook{2}(:,1:2,12)=W(:,[3,4],2); Codebook{2}(:,1:2,13)=W(:,[1,4],3); Codebook{2}(:,1:2,14)=W(:,[2,3],3); Codebook{2}(:,1:2,15)=W(:,[1,2],4); Codebook{2}(:,1:2,16)=W(:,[1,4],4);

4-Bit Rank 3 Codebook:

Codebook{3}(:,1:3,1)=W(:,[1,2,3],1); Codebook{3}(:,1:3,2)=W(:,[1,2,4],1); Codebook{3}(:,1:3,3)=W(:,[1,3,4],1); Codebook{3}(:,1:3,4)=W(:,[2,3,4],1); Codebook{3}(:,1:3,5)=W(:,[1,2,3],2); Codebook{3}(:,1:3,6)=W(:,[1,2,4],2); Codebook{3}(:,1:3,7)=W(:,[1,3,4],2); Codebook{3}(:,1:3,8)=W(:,[2,3,4],2); Codebook{3}(:,1:3,9)=W(:,[1,2,3],3); Codebook{3}(:,1:3,10)=W(:,[1,2,4],3); Codebook{3}(:,1:3,11)=W(:,[1,3,4],3); Codebook{3}(:,1:3,12)=W(:,[2,3,4],3); Codebook{3}(:,1:3,13)=W(:,[1,2,3],4); Codebook{3}(:,1:3,14)=W(:,[1,2,4],4); Codebook{3}(:,1:3,15)=W(:,[1,3,4],4); Codebook{3}(:,1:3,16)=W(:,[2,3,4],4);

4-Bit Rank Codebook:

Codebook{4}(:,1:4,1)=W(:,[1,2,3,4],1); Codebook{4}(:,1:4,2)=W(:,[1,2,3,4],2); Codebook{4}(:,1:4,3)=W(:,[1,2,3,4],3); Codebook{4}(:,1:4,4)=W(:,[1,2,3,4],4);

Specific numerical values of the 4-bit codebooks and the 6-bit codebooks disclosed in the above Table 15 and Table 16 may follow as (in this example ans(:,:,x) denotes the xth codeword):

1. 6-bit Codebooks

(1) 6-bit Rank 1 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 \\ {0.0000 + {0.5000}} \\ {{- 0.5000} + {0.0000}} \\ {{- 0.0000} - {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 \\ {{- 0.5000} + {0.0000}} \\ {0.5000 - {0.0000}} \\ {{- 0.5000} + {0.0000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 \\ {{- 0.0000} - {0.5000}} \\ {{- 0.5000} + {0.0000}} \\ {0.0000 + {0.5000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0.0000 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {{- 0.0000} - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0.0000 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {{- 0.0000} - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913}} \\ {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} \\ {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {{- 0.1913} + {0.4619}} \\ {0.4455 - {0.2270}} \\ {0.4862 + {0.1167}} \\ {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.4619} - {0.1913}} \\ {0.4455 - {0.2270}} \\ {0.1167 - {0.4862}} \\ {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} \\ {{- 0.4862} - {0.1167}} \\ {{- 0.2939} - {0.4045}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} \\ {0.4985 - {0.0392}} \\ {0.4938 - {0.0782}} \\ {{- 0.4862} + {0.1167}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} \\ {0.0392 + {0.4985}} \\ {{- 0.4938} + {0.0782}} \\ {0.1167 + {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.4985} + {0.0392}} \\ {0.4938 - {0.0782}} \\ {0.4862 - {0.1167}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.0392} - {0.4985}} \\ {{- 0.4938} + {0.0782}} \\ {{- 0.1167} - {0.4862}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},17} \right)} = \begin{matrix} 0.5000 \\ {0.4619 + {0.1913}} \\ {0.3536 + {0.3536}} \\ {0.1913 + {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} 0.5000 \\ {{- 0.1913} + {0.4619}} \\ {{- 0.3536} - {0.3536}} \\ {0.4619 - {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000` \\ {{- 0.4619} - {0.1913}} \\ {0.3536 + {0.3536}} \\ {{- 0.1913} - {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} 0.5000 \\ {0.1913 - {0.4619}} \\ {{- 0.3536} - {0.3536}} \\ {{- 0.4619} + {0.1913}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} 0.5000 \\ {0.1913 + {0.4619}} \\ {{- 0.3536} + {0.3536}} \\ {{- 0.4619} - {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 \\ {{- 0.4619} + {0.1913}} \\ {0.3536 - {0.3536}} \\ {{- 0.1913} + {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} 0.5000` \\ {{- 0.1913} - {0.4619}} \\ {{- 0.3536} + {0.3536}} \\ {0.4619 + {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} 0.5000 \\ {0.4619 - {0.1913}} \\ {0.3536 - {0.3536}} \\ {0.1913 - {0.4619}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {0.2437 + {0.4837}} \\ {0.5740 - {0.3102}} \\ {0.2823 - {0.3783}} \\ {0.2062 + {0.1248}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {0.4258 + {0.0076}} \\ {0.0596 + {0.3222}} \\ {{- 0.4186} + {0.1136}} \\ {0.1513 + {0.7073}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.5210 - {0.4265}} \\ {0.2922 + {0.0454}} \\ {0.3628 + {0.1522}} \\ {{- 0.5498} - {0.0464}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {{- 0.2364} - {0.1268}} \\ {0.6147 + {0.0409}} \\ {{- 0.6500} - {0.1085}} \\ {{- 0.2340} - {0.2440}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0591 - {0.4953}} \\ {0.5545 + {0.3908}} \\ {0.2593 - {0.0234}} \\ {0.3300 + {0.3380}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {0.0742 + {0.2714}} \\ {{- 0.0591} - {0.0963}} \\ {{- 0.5800} - {0.1645}} \\ {0.4619 + {0.5755}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.3370 + {0.2000}} \\ {0.2158 + {0.4089}} \\ {{- 0.2363} - {0.5965}} \\ {0.0175 - {0.4698}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {{- 0.1041} - {0.7125}} \\ {{- 0.5589} + {0.0299}} \\ {{- 0.2149} - {0.3331}} \\ {0.0748 - {0.0748}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5019 - {0.2171}} \\ {0.0570 + {0.3132}} \\ {0.2569 - {0.1797}} \\ {0.6949 + {0.1358}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.2121 - {0.4391}} \\ {0.3637 + {0.1907}} \\ {0.0539 - {0.3157}} \\ {{- 0.5835} + {0.3879}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.4649 + {0.1545}} \\ {0.5971 - {0.5122}} \\ {{- 0.0025} + {0.3671}} \\ {0.0058 - {0.0797}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {{- 0.4709} + {0.0383}} \\ {0.3285 + {0.0569}} \\ {0.8042 + {0.1327}} \\ {0.0341 + {0.0124}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.2199 + {0.3199}} \\ {0.5983 + {0.1264}} \\ {{- 0.5576} - {0.2903}} \\ {{- 0.0940} - {0.2672}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.6305 - {0.4345}} \\ {{- 0.0683} + {0.2985}} \\ {{- 0.0495} - {0.3235}} \\ {{- 0.0481} + {0.4588}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {{- 0.0525} + {0.2357}} \\ {0.5833 + {0.2614}} \\ {0.3552 + {0.3750}} \\ {0.0118 + {0.5160}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {{- 0.3110} - {0.3286}} \\ {0.2969 - {0.1888}} \\ {0.2888 - {0.3841}} \\ {{- 0.6633} - {0.0260}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4534 + {0.1075}} \\ {{- 0.0163} + {0.7556}} \\ {0.0681 + {0.3202}} \\ {0.1259 + {0.2977}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.2632 + {0.4348}} \\ {0.3698 - {0.0222}} \\ {{- 0.4066} - {0.6110}} \\ {0.0347 + {0.2542}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.1838 + {0.6429}} \\ {0.0055 - {0.3112}} \\ {0.3882 + {0.2198}} \\ {0.4482 - {0.2369}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2307 + {0.1558}} \\ {0.3467 - {0.2728}} \\ {0.1705 + {0.3551}} \\ {{- 0.7319} + {0.1924}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.0534 - {0.5633}} \\ {{- 0.1472} - {0.1184}} \\ {{- 0.7445} + {0.0533}} \\ {0.2881 - {0.0638}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {{- 0.4493} + {0.3172}} \\ {{- 0.5422} + {0.3203}} \\ {{- 0.1168} - {0.4073}} \\ {0.3438 + {0.0562}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {{- 0.3346} + {0.1479}} \\ {0.2302 - {0.4770}} \\ {{- 0.1288} - {0.3642}} \\ {{- 0.1358} - {0.6465}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.4868} - {0.0811}} \\ {0.4913 + {0.2140}} \\ {{- 0.2957} - {0.1634}} \\ {{- 0.3579} + {0.4766}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.4192 + {0.3317}} \\ {{- 0.2340} + {0.0529}} \\ {0.0491 - {0.2199}} \\ {{- 0.7746} + {0.0772}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.6462 - {0.3049}} \\ {{- 0.0388} - {0.4789}} \\ {{- 0.2962} + {0.1142}} \\ {0.1089 - {0.3821}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.1765 - {0.0924}} \\ {0.6212 - {0.0425}} \\ {0.5130 + {0.5000}} \\ {{- 0.2395} + {0.0459}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.3680 + {0.1691}} \\ {{- 0.5223} + {0.2262}} \\ {0.5366 + {0.2175}} \\ {0.4196 + {0.0255}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.0848 + {0.2248}} \\ {{- 0.2010} + {0.7357}} \\ {0.3862 + {0.2270}} \\ {{- 0.3114} + {0.2509}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.5883 + {0.1906}} \\ {{- 0.0481} - {0.2343}} \\ {0.2323 + {0.5375}} \\ {0.4661 + {0.0138}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {{- 0.4550} - {0.1147}} \\ {{- 0.3923} + {0.0511}} \\ {0.1654 - {0.1078}} \\ {0.6358 + {0.4245}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {{- 0.5202} + {0.2628}} \\ {0.3107 - {0.3288}} \\ {0.6089 + {0.2161}} \\ {{- 0.1325} - {0.1438}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {0.3968 - {0.0671}} \\ {{- 0.8289} - {0.0667}} \\ {0.1770 - {0.1687}} \\ {{- 0.2814} + {0.0866}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {0.3110 + {0.1356}} \\ {{- 0.2015} + {0.0416}} \\ {{- 0.3325} - {0.1370}} \\ {0.7347 - {0.4165}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},59} \right)} = \begin{matrix} {{- 0.2168} - {0.4540}} \\ {0.1102 - {0.2871}} \\ {0.0567 - {0.7634}} \\ {{- 0.0390} - {0.2544}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {{- 0.6008} - {0.3299}} \\ {{- 0.4011} - {0.1031}} \\ {0.2785 + {0.3837}} \\ {0.3519 - {0.1000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {0.0743 - {0.7846}} \\ {0.1307 - {0.3436}} \\ {{- 0.3387} - {0.2385}} \\ {{- 0.0962} + {0.2508}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {{- 0.4063} + {0.3394}} \\ {0.2466 + {0.0922}} \\ {{- 0.4080} - {0.5055}} \\ {{- 0.4767} - {0.0347}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {{- 0.2621} + {0.0798}} \\ {0.4783 - {0.5634}} \\ {0.5212 + {0.1951}} \\ {{- 0.2418} + {0.1026}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {0.1421 + {0.0583}} \\ {{- 0.0799} - {0.4928}} \\ {{- 0.2917} + {0.1078}} \\ {{- 0.0231} - {0.7937}} \end{matrix}$

(2) 6-Bit Rank 2 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000}} \\ 0.5000 & {0.5000 - {0.0000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} \\ 0.5000 & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} \\ {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.5000} + {0.0000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.0000 + {0.5000}} & {0.0000 + {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.0000} - {0.5000}} & {{- 0.0000} - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {0.4862 + {0.1167}} & {0.1167 - {0.4862}} \\ {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},17} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.1913 + {0.4619}} & {0.4619 - {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.1913 + {0.4619}} & {{- 0.1913} - {0.4619}} \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} + {0.3536}} \\ {{- 0.4619} - {0.1913}} & {0.4619 + {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.4619} + {0.1913}} & {{- 0.1913} - {0.4619}} \\ {0.3536 - {0.3536}} & {{- 0.3536} + {0.3536}} \\ {{- 0.1913} + {0.4619}} & {0.4619 + {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.4619} + {0.1913}} & {0.4619 - {0.1913}} \\ {0.3536 - {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.1913} + {0.4619}} & {0.1913 - {0.4619}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.4258 + {0.0076}} \\ {0.5740 - {0.3102}} & {0.0596 + {0.3222}} \\ {0.2823 - {0.3783}} & {{- 0.4186} + {0.1136}} \\ {0.2062 + {0.1248}} & {0.1513 + {0.7073}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.5210 - {0.4265}} \\ {0.5740 - {0.3102}} & {0.2922 + {0.0454}} \\ {0.2823 - {0.3783}} & {0.3826 + {0.1522}} \\ {0.2062 + {0.1248}} & {{- 0.5498} - {0.0464}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {{- 0.2364} - {0.1268}} \\ {0.5740 - {0.3102}} & {0.6147 + {0.0409}} \\ {0.2823 - {0.3783}} & {{- 0.6500} - {0.1085}} \\ {0.2062 + {0.1248}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} {0.4258 + {0.0076}} & {0.5210 - {0.4265}} \\ {0.0596 + {0.3222}} & {0.2922 + {0.0454}} \\ {{- 0.4186} + {0.1136}} & {0.3628 + {0.1522}} \\ {0.1513 + {0.7073}} & {{- 0.5498} - {0.0464}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {0.4258 + {0.0076}} & {{- 0.2364} - {0.1268}} \\ {0.0596 + {0.3222}} & {0.6147 + {0.0409}} \\ {{- 0.4186} + {0.1136}} & {{- 0.6500} - {0.1085}} \\ {0.1513 + {0.7073}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {0.5210 - {0.4265}} & {{- 0.2364} - {0.1268}} \\ {0.2922 + {0.0454}} & {0.6147 + {0.0409}} \\ {0.3628 + {0.1522}} & {{- 0.6500} - {0.1085}} \\ {{- 0.5498} - {0.0464}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.0742 + {0.2714}} \\ {0.5545 + {0.3908}} & {{- 0.0591} - {0.0963}} \\ {0.2593 - {0.0234}} & {{- 0.5800} - {0.1645}} \\ {0.3300 + {0.3380}} & {0.4619 + {0.5755}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.3370 + {0.2000}} \\ {0.5545 + {0.3908}} & {0.2158 + {0.4089}} \\ {0.2593 - {0.0234}} & {{- 0.2363} - {0.5965}} \\ {0.3300 + {0.3380}} & {0.0175 - {0.4698}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {{- 0.1041} - {0.7125}} \\ {0.5545 + {0.3908}} & {{- 0.5589} + {0.0299}} \\ {0.2593 - {0.0234}} & {{- 0.2149} - {0.3331}} \\ {0.3300 + {0.3380}} & {0.0748 - {0.0748}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {0.0742 + {0.2714}} & {0.3370 + {0.2000}} \\ {{- 0.0591} - {0.0963}} & {0.2158 + {0.4089}} \\ {{- 0.5800} - {0.1645}} & {{- 0.2363} - {0.5965}} \\ {0.4619 + {0.5755}} & {0.0175 - {0.4698}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.0742 + {0.2714}} & {{- 0.1041} - {0.7125}} \\ {{- 0.0591} - {0.0963}} & {{- 0.5589} + {0.0299}} \\ {{- 0.5800} - {0.1645}} & {{- 0.2149} - {0.3331}} \\ {0.4619 + {0.5755}} & {{0.0748 - {0.0748}}\;} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.3370 + {0.2000}} & {{- 0.1041} - {0.7125}} \\ {0.2158 + {0.4089}} & {{- 0.5589} + {0.0299}} \\ {{- 0.2363} - {0.5965}} & {{- 0.2149} - {0.3331}} \\ {0.0175 - {0.4698}} & {0.0748 - {0.0748}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.2121 - {0.4391}} \\ {0.0570 + {0.3132}} & {0.3637 + {0.1907}} \\ {0.2569 - {0.1797}} & {0.0539 - {0.3157}} \\ {0.6949 + {0.1358}} & {{- 0.5835} + {0.3879}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.4649 + {0.1545}} \\ {0.0570 + {0.3132}} & {0.5971 - {0.5122}} \\ {0.2569 - {0.1797}} & {{- 0.0025} + {0.3671}} \\ {0.6949 + {0.1358}} & {0.0058 - {0.0797}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.2121 - {0.4391}} & {0.4649 + {0.1545}} \\ {0.3637 + {0.1907}} & {0.5971 - {0.5122}} \\ {0.0539 - {0.3157}} & {{- 0.0025} + {0.3671}} \\ {{- 0.5835} + {0.3879}} & {0.0058 - {0.0797}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.2121 - {0.4391}} & {{- 0.4709} + {0.0383}} \\ {0.3637 + {0.1907}} & {0.3285 + {0.0569}} \\ {0.0539 - {0.3157}} & {0.8042 + {0.1327}} \\ {{- 0.5835} + {0.3879}} & {0.0341 + {0.0124}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.2199 + {0.3199}} & {{- 0.0525} + {0.2357}} \\ {0.5983 + {0.1264}} & {0.5833 + {0.2614}} \\ {{- 0.5576} - {0.2903}} & {0.3552 + {0.3750}} \\ {{- 0.0940} - {0.2672}} & {0.0118 + {0.5160}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.6305 - {0.4345}} & {{- 0.3110} - {0.3286}} \\ {{- 0.0683} + {0.2985}} & {0.2969 - {0.1888}} \\ {{- 0.0495} - {0.3235}} & {0.2888 - {0.3841}} \\ {{- 0.0481} + {0.4588}} & {{- 0.6633} - {0.0260}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {{- 0.0525} + {0.2357}} & {{- 0.3110} - {0.3286}} \\ {0.5833 + {0.2614}} & {0.2969 - {0.1888}} \\ {0.3552 + {0.3750}} & {0.2888 - {0.3841}} \\ {0.0118 + {0.5160}} & {{- 0.6633} - {0.0260}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.2632 + {0.4348}} \\ {{- 0.0163} + {0.7556}} & {0.3698 - {0.0222}} \\ {0.0681 + {0.3202}} & {{- 0.4066} - {0.6110}} \\ {0.1259 + {0.2977}} & {0.0347 + {0.2542}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.1838 + {0.6429}} \\ {{- 0.0163} + {0.7556}} & {0.0055 - {0.3112}} \\ {0.0681 + {0.3202}} & {0.3882 + {0.2198}} \\ {0.1259 + {0.2977}} & {0.4482 - {0.2369}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.2307 + {0.1558}} \\ {{- 0.0163} + {0.7556}} & {0.3467 - {0.2728}} \\ {0.0681 + {0.3202}} & {0.1705 + {0.3551}} \\ {0.1259 + {0.2977}} & {{- 0.7319} + {0.1924}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.2632 + {0.4348}} & {0.1838 + {0.6429}} \\ {0.3698 - {0.0222}} & {0.0055 - {0.3112}} \\ {{- 0.4066} - {0.6110}} & {0.3882 + {0.2198}} \\ {0.0347 + {0.2542}} & {0.4482 - {0.2369}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2632 + {0.4348}} & {0.2307 + {0.1558}} \\ {0.3698 - {0.0222}} & {0.3467 - {0.2728}} \\ {{- 0.4066} - {0.6110}} & {0.1705 + {0.3551}} \\ {0.0347 + {0.2542}} & {0.7319 + {0.1924}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.1838 + {0.6429}} & {0.2307 + {0.1558}} \\ {0.0055 - {0.3112}} & {0.3467 - {0.2728}} \\ {0.3882 + {0.2198}} & {0.1705 + {03551}} \\ {0.4482 - {0.2369}} & {{- 0.7319} + {0.1924}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.4493} + {0.3172}} \\ {{- 0.1472} - {0.1184}} & {{- 0.5422} + {0.3203}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1168} - {0.4073}} \\ {0.2281 - {0.0638}} & {0.3438 + {0.0562}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.3346} + {0.1479}} \\ {{- 0.1472} - {0.1184}} & {0.2302 - {0.4770}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1288} - {0.3642}} \\ {0.2881 - {0.0638}} & {{- 0.1358} - {0.6465}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.4493} + {0.3172}} & {{- 0.3346} + {0.1479}} \\ {{- 0.5422} + {0.3203}} & {0.2302 - {0.4770}} \\ {{- 0.1168} - {0.4073}} & {{- 0.1288} - {0.3642}} \\ {0.3438 + {0.0562}} & {{- 0.1358} - {0.6465}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.6462 - {0.3049}} \\ {{- 0.2340} + {0.0529}} & {{- 0.0388} - {0.4789}} \\ {0.0491 - {0.2199}} & {{- 0.2962} + {0.1142}} \\ {{- 0.7746} + {0.0772}} & {0.1089 - {0.3821}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.3680 + {0.1691}} \\ {{- 0.2340} + {0.0529}} & {{- 0.5223} + {0.2262}} \\ {0.0491 - {0.2199}} & {0.5366 + {0.2175}} \\ {{- 0.7746} + {0.0772}} & {0.4196 + {0.0255}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.6462 - {0.3049}} & {0.3680 + {0.1691}} \\ {{- 0.0388} - {0.4789}} & {{- 0.5223} + {0.2262}} \\ {{- 0.2962} + {0.1142}} & {0.5366 + {0.2175}} \\ {0.1089 - {0.3821}} & {0.4196 + {0.0255}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.1765 - {0.0924}} & {0.3680 + {0.1691}} \\ {0.6212 - {0.0425}} & {{- 0.5223} + {0.2262}} \\ {0.5130 + {0.5000}} & {0.5366 + {0.2175}} \\ {{- 0.2395} + {0.0459}} & {0.4196 + {0.0255}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {0.5883 + {0.1906}} \\ {{- 0.2010} + {0.7357}} & {{- 0.0481} - {0.2343}} \\ {0.3862 + {0.2270}} & {0.2323 + {0.5375}} \\ {{- 0.3114} + {0.2509}} & {0.4661 + {0.0138}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {{- 0.4550} - {0.1147}} \\ {{- 0.2010} + {0.7357}} & {{- 0.3923} + {0.0511}} \\ {0.3862 + {0.2270}} & {0.1654 - {0.1078}} \\ {{- 0.3114} + {0.2509}} & {0.6358 + {0.4245}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {{- 0.5202} + {0.2628}} \\ {{- 0.2010} + {0.7357}} & {0.3107 - {0.3288}} \\ {0.3862 + {0.2270}} & {0.6089 + {0.2161}} \\ {{- 0.3114} + {0.2509}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {0.5883 + {0.1906}} & {{- 0.4550} - {0.1147}} \\ {{- 0.0481} - {0.2343}} & {{- 0.3923} + {0.0511}} \\ {0.2323 + {0.5375}} & {0.1654 - {0.1078}} \\ {0.4661 + {0.0138}} & {0.6358 + {0.4245}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {0.5883 + {0.1906}} & {{- 0.5202} + {0.2628}} \\ {{- 0.0481} - {0.2343}} & {0.3107 - {0.3288}} \\ {0.2323 + {0.5375}} & {0.6089 + {0.2161}} \\ {0.4661 + {0.0138}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {{- 0.4550} - {0.1147}} & {{- 0.5202} + {0.2628}} \\ {{- 0.3923} + {0.0511}} & {0.3107 - {0.3288}} \\ {0.1654 - {0.1078}} & {0.6089 + {0.2161}} \\ {0.6358 + {0.4245}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${{ans}\left( {\text{:},:,59} \right)} = \begin{matrix} {0.3968 - {0.0671}} & {{- 0.6008} - {0.3299}} \\ {{- 0.8289} - {0.0667}} & {{- 0.4011} - {0.1031}} \\ {0.1770 - {0.1687}} & {0.2785 + {0.3837}} \\ {{- 0.2814} + {0.0866}} & {0.3519 - {0.1000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {0.3110 + {0.1356}} & {{- 0.2168} - {0.4540}} \\ {{- 0.2015} + {0.0416}} & {0.1102 - {0.2871}} \\ {{- 0.3325} - {0.1370}} & {0.0567 - {0.7634}} \\ {0.7347 - {0.4165}} & {{- 0.0390} - {0.2544}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {0.0743 - {0.7846}} & {{- 0.2621} + {0.0798}} \\ {0.1307 - {0.3436}} & {0.4783 - {0.5634}} \\ {{- 0.3387} - {0.2385}} & {0.5212 + {0.1951}} \\ {{- 0.0962} + {0.2508}} & {{- 0.2418} + {0.1026}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {{- 0.4063} + {0.3394}} & {{- 0.2621} + {0.0798}} \\ {0.2466 + {0.0922}} & {0.4783 - {0.5634}} \\ {{- 0.4080} - {0.5055}} & {0.5212 + {0.1951}} \\ {{- 0.4767} - {0.0347}} & {{- 0.2418} + {0.1026}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {{- 0.4063} + {0.3394}} & {0.1421 + {0.0583}} \\ {0.2466 + {0.0922}} & {{- 0.0799} - {0.4928}} \\ {{- 0.4080} - {0.5055}} & {{- 0.2917} + {0.1078}} \\ {{- 0.4767} - {0.0347}} & {{- 0.0231} - {0.7937}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.2621} + {0.0798}} & {0.1421 + {0.0583}} \\ {0.4783 - {0.5634}} & {{- 0.0799} - {0.4928}} \\ {0.5212 + {0.1951}} & {{- 0.2917} + {0.1078}} \\ {{- 0.2418} + {0.1026}} & {{- 0.0231} - {0.7937}} \end{matrix}$

(3) 6-Bit Rank 3 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536`}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536`}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536`}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536`}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {0.1167 - {0.4862}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {0.4862 + {0.1167}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {0.4938 - {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {0.4862 - {0.1167}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {{- 0.1167} - {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {{- 0.4938} + {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {0.1167 + {0.4862}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

${{ans}\left( {\text{:},:,17} \right)} = \begin{matrix} 0.5000` & 0.5000 & 0.5000 \\ {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 + {0.3536}} \\ {0.1913 + {0.4619}} & {0.4619 - {0.1913}} & {{- 0.1913} - {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},18} \right)} = \begin{matrix} 0.5000` & 0.5000 & 0.5000 \\ {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {0.1913 - {0.4619}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.1913 + {0.4619}} & {0.4619 - {0.1913}} & {{- 0.4619} + {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},19} \right)} = \begin{matrix} 0.5000` & 0.5000 & 0.5000 \\ {0.4619 + {0.1913}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.3536 + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.1913 + {0.4619}} & {{- 0.1913} - {0.4619}} & {{- 0.4619} + {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},20} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {{- 0.3536} - {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.4619 - {0.1913}} & {{- 0.1913} - {0.4619}} & {{- 0.4619} + {0.1913}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},21} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619}} & {{- 0.4619} + {0.1913}} & {{- 0.1913} - {0.4619}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} + {0.3536}} \\ {{- 0.4619} - {0.1913}} & {{- 0.1913} + {0.4619}} & {0.4619 + {0.1913}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},22} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619}} & {{- 0.4619} + {0.1913}} & {0.4619 - {0.1913}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.4619} - {0.1913}} & {{- 0.1913} + {0.4619}} & {0.1913 - {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},23} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619}} & {{- 0.1913} - {0.4619}} & {0.4619 - {0.1913}} \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.4619} - {0.1913}} & {0.4619 + {0.1913}} & {0.1913 - {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},24} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.4619} + {0.1913}} & {{- 0.1913} - {0.4619}} & {0.4619 - {0.1913}} \\ {0.3536 - {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.1913} + {0.4619}} & {0.4619 + {0.1913}} & {0.1913 - {0.4619}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},25} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.4258 + {0.0076}} & {0.5210 - {0.4265}} \\ {0.5740 - {0.3102}} & {0.0596 + {0.3222}} & {0.2922 + {0.0454}} \\ {0.2823 - {0.3783}} & {{- 0.4186} + {0.1136}} & {0.3628 + {0.1522}} \\ {0.2062 + {0.1248}} & {0.1513 + {0.7073}} & {{- 0.5498} - {0.0464}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},26} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.4258 + {0.0076}} & {{- 0.2364} - {0.1268}} \\ {0.5740 - {0.3102}} & {0.0596 + {0.3222}} & {0.6147 + {0.0409}} \\ {0.2823 - {0.3783}} & {{- 0.4186} + {0.1136}} & {{- 0.6500} - {0.1085}} \\ {0.2062 + {0.1248}} & {0.1513 + {0.7073}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},27} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.5210 - {0.4265}} & {{- 0.2364} - {0.1268}} \\ {0.5740 - {0.3102}} & {0.2922 + {0.0454}} & {0.6147 + {0.0409}} \\ {0.2823 - {0.3783}} & {0.3628 + {0.1522}} & {{- 0.6500} - {0.1085}} \\ {0.2062 + {0.1248}} & {{- 0.5498} - {0.0464}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},28} \right)} = \begin{matrix} {0.4258 + {0.0076}} & {0.5210 - {0.4265}} & {{- 0.2364} - {0.1268}} \\ {0.0596 + {0.3222}} & {0.2922 + {0.0454}} & {0.6147 + {0.0409}} \\ {{- 0.4186} + {0.1136}} & {0.3628 + {0.1522}} & {{- 0.6500} - {0.1085}} \\ {0.1513 + {0.7073}} & {{- 0.5498} - {0.0464}} & {{- 0.2340} - {0.2440}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},29} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.0742 + {0.2714}} & {0.3370 + {0.2000}} \\ {0.5545 + {0.3908}} & {{- 0.0591} - {0.0963}} & {0.2158 + {0.4089}} \\ {0.2593 - {0.0234}} & {{- 0.5800} - {0.1645}} & {{- 0.2363} - {0.5965}} \\ {0.3300 + {0.3380}} & {0.4619 + {0.5755}} & {0.0175 - {0.4698}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},30} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.0742 + {0.2714}} & {{- 0.1041} - {0.7125}} \\ {0.5545 + {0.3908}} & {{- 0.0591} - {0.0963}} & {{- 0.5589} + {0.0299}} \\ {0.2593 - {0.0234}} & {{- 0.5800} - {0.1645}} & {{- 0.2149} - {0.3331}} \\ {0.3300 + {0.3380}} & {0.4619 + {0.5755}} & {0.0748 - {0.0748}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},31} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.3370 + {0.2000}} & {{- 0.1041} - {0.7125}} \\ {0.5545 + {0.3908}} & {0.2158 + {0.4089}} & {{- 0.5589} + {0.0299}} \\ {0.2593 - {0.0234}} & {{- 0.2363} - {0.5965}} & {{- 0.2149} - {0.3331}} \\ {0.3300 + {0.3380}} & {0.0175 - {0.4698}} & {0.0748 - {0.0748}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},32} \right)} = \begin{matrix} {0.0742 + {0.2714}} & {0.3370 + {0.2000}} & {{- 0.1041} - {0.7125}} \\ {{- 0.0591} - {0.0963}} & {0.2158 + {0.4089}} & {{- 0.5589} + {0.0299}} \\ {{- 0.5800} - {0.1645}} & {{- 0.2363} - {0.5965}} & {{- 0.2149} - {0.3331}} \\ {0.4619 + {0.5755}} & {0.0175 - {0.4698}} & {0.0748 - {0.0748}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},33} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.2121 - {0.4391}} & {0.4649 + {0.1545}} \\ {0.0570 + {0.3132}} & {0.3637 + {0.1907}} & {0.5971 - {0.5122}} \\ {0.2569 - {0.1797}} & {0.0539 - {0.3157}} & {{- 0.0025} + {0.3671}} \\ {0.6949 + {0.1358}} & {{- 0.5835} + {0.3879}} & {0.0058 - {0.0797}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},34} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.2121 - {0.4391}} & {{- 0.4709} + {0.0383}} \\ {0.0570 + {0.3132}} & {0.3637 + {0.1907}} & {0.3285 + {0.0569}} \\ {0.2569 - {0.1797}} & {0.0539 - {0.3157}} & {0.8042 + {0.1327}} \\ {0.6949 + {0.1358}} & {{- 0.5835} + {0.3879}} & {0.0341 + {0.0124}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},35} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.4649 + {0.1545}} & {{- 0.4709} + {0.0383}} \\ {0.0570 + {0.3132}} & {0.5971 - {0.5122}} & {0.3285 + {0.0569}} \\ {0.2569 - {0.1797}} & {{- 0.0025} + {0.3671}} & {0.8042 + {0.1327}} \\ {0.6949 + {0.1358}} & {0.0058 - {0.0797}} & {0.0341 + {0.0124}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},36} \right)} = \begin{matrix} {0.2121 - {0.4391}} & {0.4649 + {0.1545}} & {{- 0.4709} + {0.0383}} \\ {0.3637 + {0.1907}} & {0.5971 - {0.5122}} & {0.3285 + {0.0569}} \\ {0.0539 - {0.3157}} & {{- 0.0025} + {0.3671}} & {0.8042 + {0.1327}} \\ {{- 0.5835} + {0.3879}} & {0.0058 - {0.0797}} & {0.0341 + {0.0124}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},37} \right)} = \begin{matrix} {0.2199 + {0.3199}} & {0.6305 - {0.4345}} & {{- 0.0525} + {0.2357}} \\ {0.5983 + {0.1264}} & {{- 0.0683} + {0.2985}} & {0.5833 + {0.2614}} \\ {{- 0.5576} - {0.2903}} & {{- 0.0495} - {0.3235}} & {0.3552 + {0.3750}} \\ {{- 0.0940} - {0.2672}} & {{- 0.0481} + {0.4588}} & {0.0118 + {0.5160}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},38} \right)} = \begin{matrix} {0.2199 + {0.3199}} & {0.6305 - {0.4345}} & {{- 0.3110} - {0.3286}} \\ {0.5983 + {0.1264}} & {{- 0.0683} + {0.2985}} & {0.2969 - {0.1888}} \\ {{- 0.5576} - {0.2903}} & {{- 0.0495} - {0.3235}} & {0.2888 - {0.3841}} \\ {{- 0.0940} - {0.2672}} & {{- 0.0481} + {0.4588}} & {{- 0.6633} - {0.0260}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},39} \right)} = \begin{matrix} {0.2199 + {0.3199}} & {{- 0.0525} + {0.2357}} & {{- 0.3110} - {0.3286}} \\ {0.5983 + {0.1264}} & {0.5833 + {0.2614}} & {0.2969 - {0.1888}} \\ {{- 0.5576} - {0.2903}} & {0.3552 + {0.3750}} & {0.2888 - {0.3841}} \\ {{- 0.0940} - {0.2672}} & {0.0118 + {0.5160}} & {{- 0.6633} - {0.0260}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},40} \right)} = \begin{matrix} {0.6305 - {0.4345}} & {{- 0.0525} + {0.2357}} & {{- 0.3110} - {0.3286}} \\ {{- 0.0683} + {0.2985}} & {0.5833 + {0.2614}} & {0.2969 - {0.1888}} \\ {{- 0.0495} - {0.3235}} & {0.3552 + {0.3750}} & {0.2888 - {0.3841}} \\ {{- 0.0481} + {0.4588}} & {0.0118 + {0.5160}} & {{- 0.6633} - {0.0260}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},41} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.2632 + {0.4348}} & {0.1838 + {0.6429}} \\ {{- 0.0163} + {0.7556}} & {0.3698 - {0.0222}} & {0.0055 - {0.3112}} \\ {0.0681 + {0.3202}} & {{- 0.4066} - {0.6110}} & {0.3882 + {0.2198}} \\ {0.1259 + {0.2977}} & {0.0347 + {0.2542}} & {0.4482 - {0.2369}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},42} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.2632 + {0.4348}} & {0.2307 + {0.1558}} \\ {{- 0.0163} + {0.7556}} & {0.3698 - {0.0222}} & {0.3467 - {0.2728}} \\ {0.0681 + {0.3202}} & {{- 0.4066} - {0.6110}} & {0.1705 + {0.3551}} \\ {0.1259 + {0.2977}} & {0.0347 + {0.2542}} & {{- 0.7319} + {0.1924}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},43} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.1838 + {0.6429}} & {0.2307 + {0.1558}} \\ {{- 0.0163} + {0.7556}} & {0.0055 - {0.3112}} & {0.3467 - {0.2728}} \\ {0.0681 + {0.3202}} & {0.3882 + {0.2198}} & {0.1705 + {0.3551}} \\ {0.1259 + {0.2977}} & {0.4482 - {0.2369}} & {{- 0.7319} + {0.1924}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},44} \right)} = \begin{matrix} {0.2632 + {0.4348}} & {0.1838 + {0.6429}} & {0.2307 + {0.1558}} \\ {0.3698 - {0.0222}} & {0.0055 - {0.3112}} & {0.3467 - {0.2728}} \\ {{- 0.4066} - {0.6110}} & {0.3882 + {0.2198}} & {0.1705 + {0.3551}} \\ {0.0347 + {0.2542}} & {0.4482 - {0.2369}} & {{- 0.7319} + {0.1924}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},45} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.4493} + {0.3172}} & {{- 0.3346} + {0.1479`}} \\ {{- 0.1472} - {0.1184}} & {{- 0.5422} + {0.3203}} & {0.2302 - {0.4770}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1168} - {0.4073}} & {{- 0.1288} - {0.3642}} \\ {0.2881 - {0.0638}} & {0.3438 + {0.0562}} & {{- 0.1358} - {0.6465}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},46} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.4493} + {0.3172}} & {{- 0.4868} - {0.0811}} \\ {{- 0.1472} - {0.1184}} & {{- 0.5422} + {0.3203}} & {0.4913 + {0.2140}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1168} - {0.4073}} & {{- 0.2957} - {0.1634}} \\ {0.2881 - {0.0638}} & {0.3438 + {0.0562}} & {{- 0.3579} + {0.4766}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},47} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.3346} + {0.1479}} & {{- 0.4868} - {0.0811}} \\ {{- 0.1472} - {0.1184}} & {0.2302 - {0.4770}} & {0.4913 + {0.2140}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1288} - {0.3642}} & {{- 0.2957} - {0.1634}} \\ {0.2881 - {0.0638}} & {{- 0.1358} - {0.6465}} & {{- 0.3579} + {0.4766}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},48} \right)} = \begin{matrix} {{- 0.4493} + {0.3172}} & {{- 0.3346} + {0.1479}} & {{- 0.4868} - {0.0811}} \\ {{- 0.5422} + {0.3203}} & {0.2302 - {0.4770}} & {0.4913 + {0.2140}} \\ {{- 0.1168} - {0.4073}} & {{- 0.1288} - {0.3642`}} & {{- 0.2957} - {0.1634}} \\ {0.3438 + {0.0562}} & {{- 0.1358} - {0.6465}} & {{- 0.3579} + {0.4766}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},49} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.6462 - {0.3049}} & {0.1765 - {0.0924}} \\ {{- 0.2340} + {0.0529}} & {{- 0.0388} - {0.4789}} & {0.6212 - {0.0425}} \\ {0.0491 - {0.2199}} & {{- 0.2962} + {0.1142}} & {0.5130 + {0.5000}} \\ {{- 0.7746} + {0.0772}} & {0.1089 - {0.3821}} & {{- 0.2395} + {0.0459}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},50} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.6462 - {0.3049}} & {0.3680 + {0.1691}} \\ {{- 0.2340} + {0.0529}} & {{- 0.0388} - {0.4789}} & {{- 0.5223} + {0.2262}} \\ {0.0491 - {0.2199}} & {{- 0.2962} + {0.1142}} & {0.5366 + {0.2175}} \\ {{- 0.7746} + {0.0772}} & {0.1089 - {0.3821}} & {0.4196 + {0.0255}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},51} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.1765 - {0.0924}} & {0.3680 + {0.1691}} \\ {{- 0.2340} + {0.0529}} & {0.6212 - {0.0425}} & {{- 0.5223} + {0.2262}} \\ {0.0491 - {0.2199}} & {0.5130 + {0.5000}} & {0.5366 + {0.2175}} \\ {{- 0.7746} + {0.0772}} & {{- 0.2395} + {0.0459}} & {0.4196 + {0.0255}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},52} \right)} = \begin{matrix} {0.6462 - {0.3049}} & {0.1765 - {0.0924}} & {0.3680 + {0.1691}} \\ {{- 0.0388} - {0.4789}} & {0.6212 - {0.0425}} & {{- 0.5223} + {0.2262}} \\ {{- 0.2962} + {0.1142}} & {0.5130 + {0.5000}} & {0.5366 + {0.2175}} \\ {0.1089 - {0.3821}} & {{- 0.2395} + {0.0459}} & {0.4196 + {0.0255}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},53} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {0.5883 + {0.1906}} & {{- 0.4550} - {0.1147}} \\ {{- 0.2010} + {0.7357}} & {{- 0.0481} - {0.2343`}} & {{- 0.3923} + {0.0511}} \\ {0.3862 + {0.2270}} & {0.2323 + {0.5375}} & {0.1654 - {0.1078}} \\ {{- 0.3114} + {0.2509}} & {0.4661 + {0.0138}} & {0.6358 + {0.4245}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},54} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {0.5883 + {0.1906}} & {{- 0.5202} + {0.2628}} \\ {{- 0.2010} + {0.7357}} & {{- 0.0481} - {0.2343}} & {0.3107 - {0.3288}} \\ {0.3862 + {0.2270}} & {0.2323 + {0.5375}} & {0.6089 + {0.2161}} \\ {{- 0.3114} + {0.2509}} & {0.4661 + {0.0138}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},55} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {{- 0.4550} - {0.1147}} & {{- 0.5202} + {0.2628}} \\ {{- 0.2010} + {0.7357}} & {{- 0.3923} + {0.0511}} & {0.3107 - {0.3288}} \\ {0.3862 + {0.2270`}} & {0.1654 - {0.1078}} & {0.6089 + {0.2161}} \\ {{- 0.3114} + {0.2509}} & {0.6358 + {0.4245}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},56} \right)} = \begin{matrix} {0.5883 + {0.1906}} & {{- 0.4550} - {0.1147}} & {{- 0.5202} + {0.2628}} \\ {{- 0.0481} - {0.2343}} & {{- 0.3923} + {0.0511}} & {0.3107 - {0.3288}} \\ {0.2323 + {0.5375}} & {0.1654 - {0.1078}} & {0.6089 + {0.2161}} \\ {0.4661 + {0.0138}} & {0.6358 + {0.4245}} & {{- 0.1325} - {0.1438}} \end{matrix}$

$\; {{{ans}\left( {\text{:},\text{:},57} \right)} = \begin{matrix} {0.3968 - {0.0671}} & {0.3110 + {0.1356}} & {{- 0.2168} - {0.4540}} \\ {{- 0.8289} - {0.0667}} & {{- 0.2015} + {0.0416}} & {0.1102 - {0.2871}} \\ {0.1770 - {0.1687}} & {{- 0.3325} - {0.1370}} & {0.0567 - {0.7634}} \\ {{- 0.2814} + {0.0866}} & {0.7347 - {0.4165}} & {{- 0.0390} - {0.2544}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},58} \right)} = \begin{matrix} {0.3968 - {0.0671`}} & {0.3110 + {0.1356}} & {{- 0.6008} - {0.3299}} \\ {{- 0.8289} - {0.0667}} & {{- 0.2015} + {0.0416}} & {{- 0.4011} - {0.1031}} \\ {0.1770 - {0.1687}} & {{- 0.3325} - {0.1370}} & {0.2785 + {0.3837}} \\ {{- 0.2814} + {0.0866}} & {0.7347 - {0.4165}} & {0.3519 - {0.1000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},59} \right)} = \begin{matrix} {0.3968 - {0.0671}} & {{- 0.2168} - {0.4540}} & {{- 0.6008} - {0.3299}} \\ {{- 0.8289} - {0.0667}} & {0.1102 - {0.2871}} & {{- 0.4011} - {0.1031}} \\ {0.1770 - {0.1687}} & {0.0567 - {0.7634}} & {0.2785 + {0.3837`}} \\ {{- 0.2814} + {0.0866}} & {{- 0.0390} - {0.2544}} & {0.3519 - {0.1000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},60} \right)} = \begin{matrix} {0.3110 + {0.1356}} & {{- 0.2168} - {0.4540}} & {{- 0.6008} - {0.3299}} \\ {{- 0.2015} + {0.0416}} & {0.1102 - {0.2871}} & {{- 0.4011} - {0.1031}} \\ {{- 0.3325} - {0.1370}} & {0.0567 - {0.7634}} & {0.2785 + {0.3837}} \\ {0.7347 - {0.4165}} & {{- 0.0390} - {0.2544}} & {0.3519 - {0.1000}} \end{matrix}$

$\; {{{ans}\left( {\text{:},\text{:},61} \right)} = \begin{matrix} {0.0743 - {0.7846}} & {{- 0.4063} + {0.3394}} & {{- 0.2621} + {0.0798}} \\ {0.1307 - {0.3436}} & {0.2466 + {0.0922}} & {0.4783 - {0.5634}} \\ {{- 0.3387} - {0.2385}} & {{- 0.4080} - {0.5055}} & {0.5212 + {0.1951}} \\ {{- 0.0962} + {0.2508}} & {{- 0.4767} - {0.0347}} & {{- 0.2418} + {0.1026}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},62} \right)} = \begin{matrix} {0.0743 - {0.7846}} & {{- 0.4063} + {0.3394}} & {0.1421 + {0.0583}} \\ {0.1307 - {0.3436}} & {0.2466 + {0.0922}} & {{- 0.0799} - {0.4928}} \\ {{- 0.3387} - {0.2385}} & {{- 0.4080} - {0.5055}} & {{- 0.2917} + {0.1078}} \\ {{- 0.0962} + {0.2508}} & {{- 0.4767} - {0.0347}} & {{- 0.0231} - {0.7937}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},63} \right)} = \begin{matrix} {0.0743 - {0.7846}} & {{- 0.2621} + {0.0798}} & {0.1421 + {0.0583}} \\ {0.1307 - {0.3436}} & {0.4783 - {0.5634}} & {{- 0.0799} - {0.4928}} \\ {{- 0.3387} - {0.2385}} & {0.5212 + {0.1951}} & {{- 0.2917} + {0.1078}} \\ {{- 0.0962} + {0.2508}} & {{- 0.2418} + {0.1026}} & {{- 0.0231} - {0.7937}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},64} \right)} = \begin{matrix} {{- 0.4063} + {0.3394}} & {{- 0.2621} + {0.0798}} & {0.1421 + {0.0583}} \\ {0.2466 + {0.0922}} & {0.4783 - {0.5634}} & {{- 0.0799} - {0.4928}} \\ {{- 0.4080} - {0.5055}} & {0.5212 + {0.1951}} & {{- 0.2917} + {0.1078}} \\ {{- 0.4767} - {0.0347}} & {{- 0.2418} + {0.1026}} & {{- 0.0231} - {0.7937}} \end{matrix}$

(4) 6-Bit Rank 4 Codebook:

$\; {{{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000`} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

$\; {{{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.1913 + {0.4619}} & {0.4619 - {0.1913}} & {{- 0.1913} - {0.4619}} & {{- 0.4619} + {0.1913}} \end{matrix}}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.1913 + {0.4619}} & {{- 0.4619} + {0.1913`}} & {{- 0.1913} - {0.4619}} & {0.4619 - {0.1913}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.4619} - {0.1913}} & {{- 0.1913} + {0.4619}} & {0.4619 + {0.1913}} & {0.1913 - {0.4619}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} {0.2437 + {0.4837}} & {0.4258 + {0.0076}} & {0.5210 - {0.4265}} & {{- 0.2364} - {0.1268}} \\ {0.5740 - {0.3102}} & {0.0596 + {0.3222}} & {0.2922 + {0.0454}} & {0.6147 + {0.0409}} \\ {0.2823 - {0.3783}} & {{- 0.4186} + {0.1136}} & {0.3628 + {0.1522}} & {{- 0.6500} - {0.1085}} \\ {0.2062 + {0.1248}} & {0.1513 + {0.7073}} & {{- 0.5498} - {0.0464}} & {{- 0.2340} - {0.2440}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} {0.0591 - {0.4953}} & {0.0742 + {0.2714}} & {0.3370 + {0.2000}} & {{- 0.1041} - {0.7125}} \\ {0.5545 + {0.3908}} & {{- 0.0591} - {0.0963}} & {0.2158 + {0.4089}} & {{- 0.5589} + {0.0299}} \\ {0.2593 - {0.0234}} & {{- 0.5800} - {0.1645}} & {{- 0.2363} - {0.5965}} & {{- 0.2149} - {0.3331}} \\ {0.3300 + {0.3380}} & {0.4619 + {0.5755}} & {0.0175 - {0.4698}} & {0.0748 - {0.0748}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.5019 - {0.2171}} & {0.2121 - {0.4391}} & {0.4649 + {0.1545}} & {{- 0.4709} + {0.0383}} \\ {0.0570 + {0.3132}} & {0.3637 + {0.1907}} & {0.5971 - {0.5122}} & {0.3285 + {0.0569}} \\ {0.2569 - {0.1797}} & {0.0539 - {0.3157}} & {{- 0.0025} + {0.3671}} & {0.8042 + {0.1327}} \\ {0.6949 + {0.1358}} & {{- 0.5835} + {0.3879}} & {0.0058 - {0.0797}} & {0.0341 + {0.0124}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.2199 + {0.3199}} & {0.6305 - {0.4345}} & {{- 0.0525} + {0.2357}} & {{- 0.3110} - {0.3286}} \\ {0.5983 + {0.1264}} & {{- 0.0683} + {0.2985}} & {0.5833 + {0.2614}} & {0.2969 - {0.1888}} \\ {{- 0.5576} - {0.2903}} & {{- 0.0495} - {0.3235}} & {0.3552 + {0.3750}} & {0.2888 - {0.3841}} \\ {{- 0.0940} - {0.2672}} & {{- 0.0481} + {0.4588}} & {0.0118 + {0.5160}} & {{- 0.6633} - {0.0260}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4534 + {0.1075}} & {0.2632 + {0.4348}} & {0.1838 + {0.6429}} & {0.2307 + {0.1558}} \\ {{- 0.0163} + {0.7556}} & {0.3698 - {0.0222}} & {0.0055 - {0.3112}} & {0.3467 - {0.2728}} \\ {0.0681 + {0.3202}} & {{- 0.4066} - {0.6110}} & {0.3882 + {0.2198}} & {0.1705 + {0.3551}} \\ {0.1259 + {0.2977}} & {0.0347 + {0.2542}} & {0.4482 - {0.2369}} & {{- 0.7319} + {0.1924}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.0534 - {0.5633}} & {{- 0.4493} + {0.3172}} & {{- 0.3346} + {0.1479}} & {{- 0.4868} - {0.0811}} \\ {{- 0.1472} - {0.1184}} & {{- 0.5422} + {0.3203}} & {0.2302 - {0.4770}} & {0.4913 + {0.2140}} \\ {{- 0.7445} + {0.0533}} & {{- 0.1168} - {0.4073}} & {{- 0.1288} - {0.3642}} & {{- 0.2957} - {0.1634}} \\ {0.2881 - {0.0638}} & {0.3438 + {0.0562}} & {{- 0.1358} - {0.6465}} & {{- 0.3579} + {0.4766}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4192 + {0.3317}} & {0.6462 - {0.3049}} & {0.1765 - {0.0924}} & {0.3680 + {0.1691}} \\ {{- 0.2340} + {0.0529}} & {{- 0.0388} - {0.4789}} & {0.6212 - {0.0425}} & {{- 0.5223} + {0.2262}} \\ {0.0491 - {0.2199}} & {{- 0.2962} + {0.1142}} & {0.5130 + {0.5000}} & {0.5366 + {0.2175}} \\ {{- 0.7746} + {0.0772}} & {0.1089 - {0.3821}} & {{- 0.2395} + {0.0459}} & {0.4196 + {0.0255}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {0.0848 + {0.2248}} & {0.5883 + {0.1906}} & {{- 0.4550} - {0.1147}} & {{- 0.5202} + {0.2628}} \\ {{- 0.2010} + {0.7357}} & {{- 0.0481} - {0.2343}} & {{- 0.3923} + {0.0511}} & {0.3107 - {0.3288}} \\ {0.3862 + {0.2270}} & {0.2323 + {0.5375}} & {0.1654 - {0.1078}} & {0.6089 + {0.2161}} \\ {{- 0.3114} + {0.2509}} & {0.4661 + {0.0138}} & {0.6358 + {0.4245}} & {{- 0.1325} - {0.1438}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {0.3968 - {0.0671}} & {0.3110 + {0.1356}} & {{- 0.2168} - {0.4540}} & {{- 0.6008} - {0.3299}} \\ {{- 0.8289} - {0.0667}} & {{- 0.2015} + {0.0416}} & {0.1102 - {0.2871}} & {{- 0.4011} - {0.1031}} \\ {0.1770 - {0.1687}} & {{- 0.3325} - {0.1370}} & {0.0567 - {0.7634}} & {0.2785 + {0.3837}} \\ {{- 0.2814} + {0.0866}} & {0.7347 - {0.4165}} & {{- 0.0390} - {0.2544}} & {0.3519 - {0.1000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {0.0743 - {0.7846}} & {{- 0.4063} + {0.3394}} & {{- 0.2621} + {0.0798}} & {0.1421 + {0.1583}} \\ {0.1307 - {0.3436}} & {0.2466 + {0.0922}} & {0.4783 - {0.5634}} & {{- 0.0799} - {0.4928}} \\ {{- 0.3387} - {0.2385}} & {{- 0.4080} - {0.5055}} & {0.5212 + {0.1951}} & {{- 0.2917} + {0.1078}} \\ {{- 0.0962} + {0.2508}} & {{- 0.4767} - {0.0347}} & {{- 0.2418} + {0.1026}} & {{- 0.0231} - {0.7937}} \end{matrix}$

2. 4-Bit Codebook

(1) 4-bit Rank 1 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 \\ 0.5000 \\ 0.5000 \\ 0.5000 \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 \\ {0.0000 + {0.5000}} \\ {{- 0.5000} + {0.0000}} \\ {{- 0.0000} - {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 \\ {{- 0.5000} + {0.0000}} \\ {0.5000 - {0.0000}} \\ {{- 0.5000} + {0.0000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 \\ {{- 0.0000} - {0.5000}} \\ {{- 0.5000} + {0.0000}} \\ {0.0000 + {0.5000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 \\ {0.3536 + {0.3536}} \\ {0.0000 + {0.5000}} \\ {{- 0.3536} + {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} + {0.3536}} \\ {{- 0.0000} - {0.5000}} \\ {0.3536 + {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 \\ {{- 0.3536} - {0.3536}} \\ {0.0000 + {0.5000}} \\ {0.3536 - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 \\ {0.3536 - {0.3536}} \\ {{- 0.0000} - {0.5000}} \\ {{- 0.3536} - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913}} \\ {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} \\ {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {{- 0.1913} + {0.4619}} \\ {0.4455 - {0.2270}} \\ {0.4862 + {0.1167}} \\ {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {{- 0.4619} - {0.1913}} \\ {0.4455 - {0.2270}} \\ {0.1167 - {0.4862}} \\ {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} \\ {{- 0.4862} - {0.1167}} \\ {{- 0.2939} - {0.4045}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} \\ {0.4985 - {0.0392}} \\ {0.4938 - {0.0782}} \\ {{- 0.4862} + {0.1167}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} \\ {0.0392 + {0.4985}} \\ {{- 0.4938} + {0.0782}} \\ {0.1167 + {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.4985} + {0.0392}} \\ {0.4938 - {0.0782}} \\ {0.4862 - {0.1167}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} \\ {{- 0.0392} - {0.4985}} \\ {{- 0.4938} + {0.0782}} \\ {{- 0.1167} - {0.4862}} \end{matrix}$

(2) 4-Bit Rank 2 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000}} \\ 0.5000 & {0.5000 - {0.0000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ 0.5000 & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} \\ 0.5000 & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} \\ {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.5000} + {0.0000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},6} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},7} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},8} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.0000 + {0.5000}} & {0.0000 + {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.0000} - {0.5000}} & {{- 0.0000} - {0.5000}} \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} 0.5000 & 0.5000 \\ {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {0.4862 + {0.1167}} & {0.1167 - {0.4862}} \\ {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

(3) 4-Bit Rank 3 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 4} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},5} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 6} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 7} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:}, 8} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 \\ {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},9} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {0.1167 - {0.4862}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},10} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},11} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},12} \right)} = \begin{matrix} {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {0.4862 + {0.1167}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$

${{ans}\left( {\text{:},\text{:},13} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {0.4938 - {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {0.4862 - {0.1167}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},14} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {{- 0.1167} - {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},15} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},16} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {{- 0.4938} - {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {0.1167 + {0.4862}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

(4) 4-Bit Rank 4 Codebook:

${{ans}\left( {\text{:},\text{:},1} \right)} = \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ 0.5000 & {0.0000 + {0.5000}} & {{- 0.5000} + {0.0000}} & {{- 0.0000} - {0.5000}} \\ 0.5000 & {{- 0.5000} + {0.0000}} & {0.5000 - {0.0000}} & {{- 0.5000} + {0.0000}} \\ 0.5000 & {{- 0.0000} - {0.5000}} & {{- 0.5000} + {0.0000}} & {0.0000 + {0.5000}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},2} \right)} =  \begin{matrix} 0.5000 & 0.5000 & 0.5000 & 0.5000 \\ {0.3536 + {0.3536}} & {{- 0.3536} + {0.3536}} & {{- 0.3536} - {0.3536}} & {0.3536 - {0.3536}} \\ {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} & {0.0000 + {0.5000}} & {{- 0.0000} - {0.5000}} \\ {{- 0.3536} + {0.3536}} & {0.3536 + {0.3536}} & {0.3536 - {0.3536}} & {{- 0.3536} - {0.3536}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},3} \right)} = \begin{matrix} {0.4619 + {0.1913}} & {{- 0.1913} + {0.4619}} & {{- 0.4619} - {0.1913}} & {0.1913 - {0.4619}} \\ {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} & {0.4455 - {0.2270}} \\ {{- 0.1167} + {0.4862}} & {0.4862 + {0.1167}} & {0.1167 - {0.4862}} & {{- 0.4862} - {0.1167}} \\ {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} & {0.2939 + {0.4045}} & {{- 0.2939} - {0.4045}} \end{matrix}$ ${{ans}\left( {\text{:},\text{:},4} \right)} = \begin{matrix} {- 0.5000} & {- 0.5000} & {- 0.5000} & {- 0.5000} \\ {0.4985 - {0.0392}} & {0.0392 + {0.4985}} & {{- 0.4985} + {0.0392}} & {{- 0.0392} - {0.4985}} \\ {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} & {0.4938 - {0.0782}} & {{- 0.4938} + {0.0782}} \\ {{- 0.4862} + {0.1167}} & {0.1167 + {0.4862}} & {0.4862 - {0.1167}} & {{- 0.1167} - {0.4862}} \end{matrix}$

FIG. 3 is a flowchart illustrating an exemplary MIMO communication method. The exemplary method may be performed by a base station and/or a terminal in a MIMO communication network. In 310, the method comprises storing a codebook in memory. In 320, the method recognizes a channel state of the MIMO communication network. In 330, the method determines a transmission rank. In 340, the method comprises determining a precoding matrix. And in 350, the method comprises performing precoding.

The methods described above may be recorded, stored, or fixed in one or more computer-readable storage media that includes program instructions to be implemented by a computer to cause a processor to execute or perform the program instructions. The media may also include, alone or in combination with the program instructions, data files, data structures, and the like. The media and program instructions may be those specially designed and constructed, or they may be of the kind well-known and available to those having skill in the computer software arts. Examples of computer-readable media include magnetic media such as hard disks, floppy disks, and magnetic tape; optical media such as CD ROM disks and DVD; magneto-optical media such as optical disks; and hardware devices that are specially configured to store and perform program instructions, such as read-only memory (ROM), random access memory (RAM), flash memory, and the like. Examples of program instructions include both machine code, such as produced by a compiler, and files containing higher level code that may be executed by the computer using an interpreter. The described hardware devices may be configured to act as one or more software modules in order to perform the operations and methods described above, or vice versa. In addition, a computer-readable storage medium may be distributed among computer systems connected through a network and computer-readable instructions or codes may be stored and executed in a decentralized manner.

FIG. 4 illustrates a configuration of a base station 410 and a terminal 420.

The base station 410 includes a memory 411, a processor 412, and a precoder 413. The terminal 420 includes a memory 421, a channel estimator 422, a processor 423, and a feedback unit 424.

The aforementioned codebooks may be stored in the memory 411 of the base station 410 and/or the memory 421 of the terminal 420. For example, any of the codebooks described herein may be stored in the memory 411 of the base station 410 and/or the memory 421 of the terminal 420.

The channel estimator 422 may estimate a channel between the base station 410 and the terminal 420. The processor 423 may select a preferred codeword matrix from the codebook stored in the memory 421, based on the estimated channel, and generate feedback data associated with the preferred codeword matrix. The feedback unit 424 may feed back the feedback data to the base station 410.

The base station 410 may receive the feedback data. The processor 412 may verify the preferred codeword matrix using the codebook stored in the memory 411. The precoder 413 may precode at least one data stream using the determined precoding matrix.

A number of exemplary embodiments have been described above. Nevertheless, it will be understood that various modifications may be made. For example, suitable results may be achieved if the described techniques are performed in a different order and/or if components in a described system, architecture, device, or circuit are combined in a different manner and/or replaced or supplemented by other components or their equivalents. Accordingly, other implementations are within the scope of the following claims. 

1. A multiple input multiple output (MIMO) communication system, comprising: a terminal to feed back feedback data using a codebook; and a base station to access a memory storing the codebook, and to precode a data stream that the base station desires to transmit using the codebook. 